Signal Processing for In-Situ Detection of Effective Heat Pulse Probe Spacing Radius as the Basis of a Self-Calibrating Heat Pulse Probe

A sensor comprised of an electronic circuit and a hybrid single and dual heat pulse probe was constructed and tested along with a novel signal processing procedure to determine changes in the effective dual-probe spacing radius over the time of measurement. The circuit utilized a proportional–integral–derivative (PID) controller to control heat inputs into the soil 10 medium in lieu of a variable resistor. The system was designed for on-board signal processing and implemented USB, RS232 and SDI-12 interfaces for Machine-to-Machine (M2M) exchange of data, thereby enabling heat inputs to be adjusted to soil conditions and data availability shortly after the time of experiment. Signal processing was introduced to provide a simplified single-probe model to determine thermal conductivity instead of reliance on late-time logarithmic curve-fitting. Homomorphic and derivative filters were used with a dual-probe model to detect changes in the effective probe spacing radius 15 over the time of experiment to compensate for physical changes in radius as well as model and experimental error. Theoretical constraints were developed for an efficient inverse of the exponential integral on an embedded system. Application of the signal processing to experiments on sand and peat improved the estimates of soil water content and bulk density compared to methods of curve-fitting nominally used for heat pulse probe experiments. Applications of the technology may be especially useful for soil and environmental conditions where effective changes in probe spacing radius need to be detected and 20 compensated over the time of experiment.

HPPs can be broadly classified into two different types: Single Probe (SP) (De Vries, 1952;Li et al., 2016) and Dual Probe (DP) Campbell et al., 1991;Ham and Benson, 2004) devices. The SP consists of a single heater 40 needle that is inserted into the geomaterial. A temperature measurement sensor (i.e. a thermistor) placed inside of the heater needle is used to determine the change in temperature as the needle releases thermal energy. The DP consists of two needles that are inserted into the geomaterial: one of the needles functions as a heater, whereas the other needle measures the change in temperature of the geomaterial at an offset distance to the heated needle. The DP has an advantage over the SP since the SP can only be used to determine thermal conductivity, whereas the DP can be used to determine thermal conductivity and 45 diffusivity of the geomaterial (Bristow et al., 1994).
An assumption nominally made in conjunction with DP sensors is that the radius is constant during each measurement.
If the DP radius changes after the probe is inserted into the geomaterial or over the time of measurement due to heating and cooling, HPP determination of thermal properties will be inaccurate (Kluitenberg et al., 1993;Mori et al., 2003). The measured thermal conductivity is not sensitive to changes in DP probe spacing radius, whereas the HPP-determined heat capacity and 50 thermal diffusivity exhibit high sensitivity to radius changes (Kluitenberg et al., 2010;Liu et al., 2007). This creates challenges in estimating the moisture content of frozen soils where thawing and freezing occur, and has required recalibration of individual probes (Zhang et al., 2011).
Most HPP researchers utilized commercially available off-the-shelf (COTS) hardware (i.e. a datalogger) to collect data (Bristow, 1998;Bristow et al., 1994Bristow et al., , 2001Kamai et al., 2008;Li et al., 2016), although recently custom electronic circuits 55 have been proposed. Valente et al. (2006) interfaced a multi-functional soil probe to a processing circuit. Dias et al. (2013) used an NPN transistor as a heat source for a SP device. The temperature of the transistor was determined using a circuit and transistor circuit theory. Sherfy et al. (2016) used an NE555 timer circuit to control the duration of the heating pulse. Miner et al. (2017) and Ravazzani (2017) developed Arduino-based HPP sensors utilizing currently-established DP theory. Liu et al. (2013), Wen et al. (2015), and Liu et al. (2016) showed that two or more thermistors placed inside the temperature measurement 60 needles of a DP device can be used to determine probe deflection. Multiple thermistors are required to determine probe deflection and the method cannot be used to calculate a time series of small changes in the probe spacing radius that occurs during the time of measurement when the heater needle increases in temperature.
A Self-Calibrating Heat Pulse Probe (SCHEPP) system is described that consists of a custom electronic circuit and novel inverse models for the SP and DP. The HPP used for SCHEPP is a hybrid of the SP and DP designs. SP and DP forward 65 models are combined and used to determine changes in the effective probe spacing radius during the time of measurement.
This effective radius compensates for model error and, similar to a calibrated probe spacing radius, does not directly coincide with the actual probe spacing radius. Another inverse model is also introduced that allows for determination of thermal conductivity without the need for an SP model late-time approximation (cf. Section 2.2.1 for rationale).

3 2 Materials and Methods
Diagrams of the SCHEPP system HPP are shown as Fig. 1a and 2a. A loop of Nichrome wire is placed inside of a heater needle, along with a measurement thermistor. Another measurement thermistor is placed inside of a temperaturesensing needle situated at an offset distance to the heater needle. Figure 2a shows that SCHEPP uses a hybrid SP and DP 75 device. A Proportional-Integral-Derivative (PID) controller is used to precisely control and maintain heat inputs in lieu of a variable resistor. Circuit theory is used to determine the resistance of the Nichrome heating wire inside of the heater needle during a measurement. This eliminates the need to use a previously measured estimate of the heater wire resistance. The heater wire resistance is directly measured over the time of an experiment. Figure 1b is a conceptual diagram that shows relationships between the models and measurement methods. 80

Forward Models
Assuming that the heater needle is an infinite line source in an infinite medium, the "late-time" change in temperature ( ) 1  t of the heater probe SP device is given by Eq. 22a of Blackwell (1954): In Eq. (1) above, q is the rate of energy transferred per unit length of the probe, k is the thermal conductivity of the geomaterial,   ,, B C D are constants and the natural logarithm is utilized. The assumption of an infinite line source in an infinite medium is valid if the heater needle has a small diameter and the geomaterial is of sufficiently large dimension to be isotropic and homogeneous throughout so that the heat pulse does not interact with dissimilar boundaries (i.e. a container in 90 which the soil is placed) over the time of the measurement (Kluitenberg et al., 1993(Kluitenberg et al., , 1995Liu et al., 2007).
where n r is the radius of the needle and  is the thermal diffusivity of the medium, the last term in Eq. (1) can be neglected (Bristow et al., 1994;Li et al., 2016). In this paper, Eq. (1) as a forward model is taken subject to the constraint that ( ) 1 0   t since negative values are not physically reasonable within the context of the model.
Assuming an infinite line source in an infinite medium for a DP device, the change in temperature ( ) 1 ,  rt sensed at 95 a radial distance r from the heater needle is (Kluitenberg et al., 1993): The thermal diffusivity of the medium is  , the density is  , and the specific heat capacity is c . The exponential integral 105 function is i E . The current through the Nichrome wire is turned on at time 0 t and turned off at time h t . Therefore, 0  h t t t is referred to as the heating period, and  h tt as the cooling period. Calibration to determine a radius using least-squares curve-fitting will yield an effective radius that is representative of differences between the sensing system and the ideal model described above. This initial radius is referred to as initial r and is taken as a constant.  (1).
Given these disadvantages, this section introduces a method, the "Signal Processing SP Model", which uses signal processing to reduce the time series associated with Eq. (1) to a simpler model. Least-squares curve-fitting is used with a modified version of Eq. (1) and the total SP dataset during heating. Errors introduced during the earlier time of heating are 120 acceptably small, particularly when signal processing has modified the SP dataset as a time-domain signal and k is obtained by a least-squares curve-fit.
Equation (1) is subjected to Hadamard (point-by-point) multiplication by t to obtain The Hadamard multiplication to produce Eq. (6) is a type of homodyning process where later-time values of ( ) 1  t are assigned greater-magnitude weights than earlier-time values. Taking the numerical time derivative is similar to application of a high-pass filter (Hamming, 1983;pg. 118). The resulting equation is: Homodyning again by t yields: Taking the numerical time derivative again: To reduce noise associated with the derivative operation when working with actual data, a Butterworth lowpass filter with zero-phase filtering and a cutoff frequency of 0.3 Hz is applied to the numerical sequence associated with Eq. (9). The cutoff 140 frequency was chosen to ensure stability of the inverse model within the context of the data used for the experiments reported by this paper. Given a known q , curve-fitting is applied to the filtered sequence associated with Eq. (9) to determine k without the need to also determine   , CD at time 2 /   n tr where the model of Eq. (1) is still valid. Once the thermal conductivity k is determined by the signal processing, curve-fitting using Eq. (1) with a starting value of the determined thermal conductivity is used to estimate parameters for application of the SP forward model. 145 The "Late-Time SP Model" is used for comparison with the Signal Processing SP Model described above. A linear section of the

Dual Probe and Variable Radius
The inverse model described in this section uses signal processing to determine ( ) rt as an effective radius that changes over the time of heating and cooling to compensate for model error and physical changes in the probe spacing. A numerical value for initial r is required as an estimate of the initial probe spacing radius. Changes in the effective probe spacing ( ) rt are 155 determined over time and used to obtain an  that is representative of these changes. This is the "Signal Processing DP" model.
Thermal conductivity, k , is determined using a measured q , and the inverse signal processing model presented in Section 2.2.1 for the SP. The thermal conductivity is not directly determined from the DP model using curve-fitting since the effective radius ( ) rt can change over time. 160 The k is used with a known q to algebraically remove the /4 qk term from Eq.
(2) to obtain Taking the time derivative of Eq. (13) is once again similar to the application of a high-pass filter that suppresses the constant ( ) 1/ 2 log 2 term. When Eq. (14) is expressed as a discrete sequence sampled at a frequency of s f the derivative is 180 approximated using a backward-difference method (Eq. (15)).
The derivative is computed using a backward difference: 185 In Eq. . For application to actual data, we assume that the probe spacing is the calibrated initial r at =+ initial p a t t t where p t is the time where the curve associated with ( ) 1 ,  rt is at a maximum and a t is an additional time delay that compensates for a non-ideal system. For this system, the additional time delay was chosen such that 0 s 2 s p t .

195
Selection of a peak time is a similar idea to the temperature maximum method (Bristow et al., 2001) where the calibrated initial r is used at the time of peak temperature change. The additional time delay a t is chosen to approximately coincide with the integer-valued time durations of moving-average windows (Section 2.5). The input data is trimmed appropriately. This selection of boundary condition is supported by tests on actual soil performed in this paper and a sensitivity analysis that justifies the selection of the additional time (16) will thereby yield a curve that is a straight line with a slope that is approximately zero when  is approximately constant over the time of heating and cooling. The ( ) rt is an effective radius that is also affected by temperature drift and deviation of the physical system from an ideal model. Since ( ) rt is an effective radius, it will not directly coincide with an actual probe spacing radius. 205

Measurement of Soil Water Content and Density 210
The heat capacity k and thermal diffusivity  are determined using an inverse model as described in the previous sections of this paper. Neglecting the contribution of air, the volumetric heat capacity of soil h C is calculated by (Kluitenberg, 2002): Re-arranging Eq. (17) and solving for volumetric water content: The  w is numerically constrained to be within the range 01 w   . The mineral content of the soil  m is known and the organic content  o can be easily determined from laboratory testing or an organic carbon soil map of a geographic area. For 9 implementation using a microcontroller,  m and  o are stored in flash (non-volatile) memory and these values change based on the geographic location of the soil. The density of the soil is determined by volume fractions: In equation (19) above, the constituent densities and heat capacities are known. The w  is determined using Equation (18). Figure 1 shows a conceptual block diagram of the system. Thermistors in half-bridge configurations are used to determine the temperatures of HPP needles.

Circuit Theory and PID Control 240
The heat input into the soil by the heater probe is: In the above Eq. (20) and Eq. (21), the electrical power is P , the total resistance of the Nichrome heater wire is w R , and is the length of the heater needle. Given a measured voltage drop E over a four-terminal Kelvin sense resistor with known 250 resistance s R , the current though the heater wire is calculated using Ohm's law: For a current I through the heater wire and sense resistor, the output voltage is measured as kn E by an analog-to-digital converter ( Fig. 1). Using Kirchhoff's voltage law for this circuit, the resistance w R of the Nichrome wire is determined at each sampling timestep by: 255 To set a constant q , a proportional-integral-derivative (PID) controller (Ang et al., 2005) is utilized. The variable voltage source is adjusted at each discrete timestep by a digital-to-analog-converter (DAC). Since E and kn E are measured 260 at each discrete timestep at a sampling rate of s f , Eq. (20) to Eq. (23) are used with the feedback loop shown in Fig. 1 to ensure that the q remains close to a set-point value during the time of experiment. The use of the PID controller requires a higher sampling rate s f than a nominal HPP experiment to adjust the output q . The PID controller thereby ensures that the soil can heat up in a controlled fashion and considers resistance changes in the Nichrome wire in lieu of using an assumed resistance. Figure 2a is a block diagram indicating how the system incorporates a microcontroller and communication 265 interfaces.

Determination of Temperature Change Curves
The sampled temperature inside the heater needle is denoted as ( )  t and the sampled temperature inside the second needle at 270 an offset distance from the heater needle is denoted as ( ) Tt. The sampled temperatures ( )  t and ( ) Tt are lowpass filtered using a 5 th order Butterworth filter with a cutoff frequency of 10 Hz applied as a zero-phase filter to reduce noise. The Butterworth filter was chosen since it is maximally-flat in the passband and the zero-phase filtering ensures that time shifts are minimized to ensure accurate application of the inverse models described in this paper using the collected data. For the DP model calibration to find an initial 0 r using curve-fitting, the sampled temperatures are processed by a 275 moving-average filter over 1 s windows to further reduce noise before curve-fitting. Alternately, for the DP inverse model (Section 2.2.2), a moving-average filter is used to obtain an equivalent sampling rate of 12 Hz to ensure that the ( ) The temperature changes are used for application of inverse models related to Eq. (1) and (2).

285
Heat inputs into the soil are determined during the time of experiment and calculated using Eq. (21). When the current is applied and travels through the heater wire, there is a short time delay ( 1 s ) before the setpoint q is attained when the heater needle increases in temperature. The interval of the time series for a constant calculated q during the time of experiment is determined using a step detection algorithm (Carter et al., 2008) based on the Student ttest (Ebdon, 1991; pp. 61-64) with a null hypothesis at a significance level of 1% and a window size of 31 elements. The 290 significance level and window size are dependent on the implementation of the sampling system and are thereby chosen to detect the step within the context of this experiment. To find an estimate of a constant value of q , the time series is averaged over the plateau of the step. The time series location of the step associated with a constant q is found by application of a sliding mean filter with a window size of 31 elements applied to a time series of electrical power P used for the computation of q using Equation (21). The mean filter is applied to a binary sequence created by mapping non-rejection of the null 295 hypothesis to binary 0 and rejection of the null hypothesis to binary 1. The plateau is coincident with a sequence of zeros away from the edges of the step. The edges of the step are indicated by non-zero elements in this sequence surrounded by zeros. The window size is appropriate for the sampling system described in the context of this paper.

Apparatus 300
A custom electronic circuit board was designed and constructed for the SCHEPP system (Fig. 2a, b). The circuit board was placed into an enclosure box and connected to the HPP by a cable and mating circular connectors. The HPP body was epoxied  (Fig. 2b). In the Analog Front End (AFE), a two-channel 24-bit ADC with a precision 10k (0.01%, ±5ppm/°C) resistor half-bridge for each channel and a 2.5V voltage reference (2 ppm/°C, ±0.02% voltage error) was used to determine the resistance of the thermistors inside of each needle. The resistance of a thermistor was related to temperature by the Steinhart-Hart equation (Steinhart and Hart, 1968 adjusted. This resistance is smaller than the 1  current sense resistor nominally used in other HPP experiments (Bristow et al., 1994;Li et al., 2016;Si, 2011, 2008;Valente et al., 2006;Zhang et al., 2011). Moreover, the precision sense resistor had a Kelvin terminal connection for precision and was physically large to reduce self-heating by current flow. The voltage drop over a sense resistor was determined by a precision difference amplifier and a 16-bit ADC, allowing for a 1 LSB step size of 2.5 V  . The output voltage kn E ( Fig. 1) was also measured by a 16-bit ADC and amplifier resulting in a 1 LSB 320 step size of 1.25mV.
A 32-bit microcontroller with a system clock of 300 MHz was used to control the HPP experiment and perform floating-point calculations (Fig. 2a). The system clock had to be set at 300 MHz to allow the microcontroller to sample all ADCs in the system at 120 Hz = s f and also perform floating point calculations associated with this application. The system clock speed is provided here to provide a starting point for engineering of similar designs. The DAC used to control the output 325 voltage was also updated at the same sampling rate with the PID controller output. The 120 Hz sampling rate enabled functioning of the PID controller feedback loop and allowed for digital filtering for signal processing.
The microcontroller had an integrated USB transceiver for communication with a computer. RS-232 and SDI-12 interfaces were also integrated into the system for communication with a computer or datalogger as Machine-to-Machine (M2M) interfaces. SDRAM stored data from the experiment and provided temporary memory for heap allocation of arrays 330 and data structures. Code for the microcontroller was written in the C programming language.
A command-line serial port interface permitted changing the duration of the experiment, set-point q value, and the time of heating. For each experiment, the microcontroller monitored the maximum temperature rise at the heater needle and terminated the experiment if the temperature rise exceeded the maximum operating temperature of the thermistors.
The mechanical construction and design of the HPP used for this paper has been reported and rationalized in other 335 papers (Li et al., 2016;Si, 2008, 2010). The needles of length 3.0 cm = were constructed from stainless-steel tubing (1.28 mm OD and 0.84 mm ID) and filled with thermally-conductive epoxy. The sense thermistor was placed in the geometric middle of each needle to prevent edge effects associated with heat conduction and the needles were filled with thermal epoxy (Saito et al., 2007). The nominal spacing between the heater needle and the sense needle was 6 mm . During laboratory testing of SCHEPP (Fig. 2c, d), the experiment was initiated by a laptop computer connected to the circuit's USB 340 port and communication was conducted over the USB interface.

345
Following Campbell et al. (1991), calibration was conducted to find initial r using a 5 g/L agar gel solution. The thermal conductivity k of the agar gel was taken to be the same as the thermal conductivity of water (Saito et al., 2007). Reported values for thermal conductivity (Ramires et al., 1995) and heat capacity (Wagner and Pruß, 2002) of water were used.
For all experiments, the temperature of the probe needles was measured for 1 s at a sampling rate of 120 Hz = s f before electrical current was applied to the Nichrome wire. This initial temperature measurement for each trial was averaged 350 over the 1 s period.
The agar gel was washed out using distilled water and the cylindrical container was packed with soil. Two types of soil were used for the HPP tests: sand and peat. These soils are indicative of the physical extent of soil thermal properties.
The soils were collected from field sites near Fort McMurray, Alberta. The sand contained small amounts of bitumen as representative of the Alberta Oil Sands area. Two independent laboratory analyses with incineration at 1100°C were conducted 355 on the sand, finding the total carbon content to range between a mean of 0.44% and 0.75% by mass. The soil properties are summarized in Table 1.
The water content for the sand was chosen so that the sand was saturated, whereas the water content for the peat was chosen so that the soil would remain as wet as possible (Table 1). Due to the absorbent characteristics of the peat soil, it was not possible during the time of the laboratory experiment to completely saturate the pore spaces of the soil column. However, 360 the volumetric water content  for both soils was chosen to ensure adequate contact between the probe and the soil and to also reduce air gaps that can increase thermal contact resistance and decrease accuracy of the measurement (Liu and Si, 2010).
These air gaps can occur in drier soils with lower water contents. Since comparisons are required to be made between heat pulse probe and gravimetric measurements for testing the in-situ calibration procedure described in this paper, the presence of air gaps represents an additional source of error that was controlled. 365 Since the soil dried out over the time of multiple experiments, some additional water was added between successive days to ensure that the volumetric water content  was close to the target value. Between trials, the top of the container was covered with a cap to reduce evaporation of water from the soil. Changes in water content occurred over the time of the experiment due to evaporation since the cap did not create a hermetic seal between the top of the container and the soil column. Table 1 shows quantities used for application of HPP forward and inverse models to peat and sand. 370 Experiment sampling durations, q heat inputs, and heat durations are summarized in Table 2. Trial numbers of each experiment refers to groups of experiments conducted temporally close together.
The heat pulse strength and time of heating were chosen to minimize interaction of the heat pulse with the container boundaries. Due to the short time span over which each experiment was conducted in a laboratory setting, explicit correction was not applied for changes in ambient temperature (Young et al., 2008;Zhang et al., 2014). Between each experiment, the 375 temperature of the soil column returned to a level that approximated the initial temperature before the probe was heated again for the next trial. All experiments were conducted at room temperature (~20°C).
Numerical comparisons were made using Root Mean Squared Difference (RMSD) and Mean Bias (MB). The RMSD indicates the overall differences between two datasets. The Mean Bias (MB) indicates whether the model under-predicts or over-predicts relative to the observations. 380 14 3. Results

Synthetic Experiments
Synthetic heating curves were constructed using Eq. (1) and Eq.
(2) to serve as a forward model and provide a test of the signal 385 processing. The SP and DP curves are shown as Fig. 3 and Fig. 4 and were generated using the model inputs given in Table   3. For the DP, the assumed change in probe spacing radius is shown for a linear increase (Fig. 4b), decrease (Fig. 4e) and Brownian random walk scaled so that the numerical values are between a starting and ending radius (Fig. 4h).
The time-variable radius is ( ) rt and the associated curve is shown on the plots as a "DP Variable Radius." The "DP Fixed Radius" curve is calculated using the first element of the ( ) rt used for a particular "DP Variable Radius" curve. The 390 fixed radius is taken to be constant over the time of heating and cooling.   Fig. 4 (a, b, c) is for a linear increase; the second row of Fig. 4 (d, e, f) is for a linear decrease; and the third row of Fig.  400 4 (g, h, i) is for the Brownian random walk. The difference between the forward and inverse models is on the order of 15 1 10 −  for all changes in probe spacing radius (Fig. 4c-i), indicating that for a synthetic model the error is mostly associated with floating point calculations and that the inverse model is accurate.

Soil Data 405
Since the "Signal Processing SP" and "Signal Processing DP" models are applied together, hereafter these will be referred to as the "Signal Processing SP and DP" model. The "Late-Time SP Model" is described in Section 2.2.1. The "Heating and Cooling DP" model refers to the nominal curve-fitting using Eq. (2). The rapid fluctuations in the effective radius ( ) rt occur due to temperature drift, model and experimental error. The signal processing thereby compensates for these effects using ( ) rt as an effective radius. Table 4 shows that the quantities found using all models have the same orders of magnitude and indicates that the Signal Processing SP and DP model is more accurate than the Heating and Cooling DP model as compared to the gravimetric values used for the laboratory experiment. Figure 6 shows the Signal Processing SP and DP model applied to peat (Table 5). Compared to the sand example 415 given above, the thermal conductivity k and diffusivity  are lower for the peat demonstrating that the peat takes longer to warm up during the time of experiment. The heating and cooling curves are thereby distinctively different between sand and peat. However, in the same manner as the sand example, the quantities found using all models are of similar orders of magnitude. Fig. 6c shows that there are fewer rapid fluctuations in the effective radius ( ) rt determined for peat as compared to sand. Moreover, the change in effective radius is less pronounced for peat as compared to sand due to smaller temperature 420 drift associated with lower k and  . Also, in a similar fashion to sand, the PD and numerical difference demonstrates that the signal processing models introduced in this paper are more accurate than the nominal models. type. The thermal diffusivity estimates provided by the Heating and Cooling DP model are slightly higher than the estimates provided by the Signal Processing SP and DP model for sand (Fig. 7c), whereas for peat (Fig. 7d) (19)) in the vicinity of the probe for experiment identifier D and demonstrates sensitivity of the signal processing to this change.
An increase in water content is apparent for experiment identifiers E to G associated with peat. The increase in water 450 content occurred since the HPP experiments were initially conducted less than an hour after water was added to the soil column.
Since the soil column was opaque, the infiltration of water in the column and the associated wetting front could not be tracked and thereby localized volumes of water surrounding the HPP caused a rise in water content. Since the rise is consistent and shown by the Heating and Cooling DP model as well as the Signal Processing SP and DP model for  w and  (Fig. 8b, d), this indicates that both models are in physical agreement. Lower RMSD, MB and PD values for the Signal Processing SP and 455 DP model indicates that the signal processing introduced in this paper also improves estimates of  w and  when the water content changes during the time interval of experiment identifier E.
For experiment identifiers F and G, the determined  w and  remains approximately constant over time for peat.
Compared to experiment identifier E, a reduction in water content has occurred due to infiltration over time and some loss of water due to evaporation. For these experiments, the RMSD, MB and PD is lower for the Signal Processing SP and DP model 460 compared to the Heating and Cooling DP model.

Sensitivity Analysis
To identify the effects of model parameters on the RMSD and MB of model outputs  w and  , a sensitivity analysis ( Fig. 9 to Fig. 10) was conducted over all data collected. The sensitivity analysis utilized the OAT (One-at-a-Time) approach, where 465 one variable at a time is changed whereas the other model inputs are held constant (Hamby, 1994). Overall, for a range of nominal model inputs, Fig. 9 to Fig. 10 demonstrate that the signal processing associated with the Signal Processing SD and DP model reduces the RMSD and MB compared to the nominal Heating and Cooling DP model. This also indicates that the signal processing method produces more accurate estimates than the curve-fitting models nominally used for heat pulse probe experiments. 470 For all models and soils used to determine  w and  , the RMSD and MB is lowest when the initial radius . The time delay a t is not a parameter for the nominal Heating and Cooling DP model and a sensitivity analysis is not conducted for a t when using this model. Applied to peat, the Signal Processing SD and DP model is relatively insensitive to the time delay a t due to the lower thermal conductivity k and diffusivity  relative to sand that dampens changes in the effective radius ( ) rt as determined by signal processing. 485 In the context of the sensitivity analysis, as  o increases for sand, the RMSD and MB related to  also increases.
For peat, a concomitant increase in  o is associated with an increase in the RMSD and MB, indicating that for  it is not possible to calibrate for  o and an approximation of  o must be known for model application within the context of these experiments. A mineral content of 0.55   m for sand produces the lowest RMSD and MB related to  . As  m increases for peat, the RMSD and MB also increases, indicating that for  it is once again not possible to calibrate for  m and an 490 approximation must be utilized.

Conclusions
• A novel circuit was designed and tested using a hybrid SP and DP Heat Pulse Probe (HPP) design. The circuit utilized a PID controller to precisely control the heat input into the soil. In lieu of a variable resistor, this enabled the heat input q to be changed by a computer or a datalogger. When deployed at a remote or inaccessible field site, the HPP 495 heat input can be set to a given value using the communication interfaces. This enables the heat input to be 18 appropriately selected for soil type.
• Instead of using a 1  sense resistor to infer heat inputs into the soil, the circuit used a resistor with a 0.01  nominal resistance. This reduced the voltage drop across the sense resistor and still allowed for the current through the 500 Nichrome wire to be adequately determined during the time of experiment, although a differential amplifier was required to detect the voltage difference before digitization by an ADC.
• A sampling rate of 120 Hz was required for application of the PID controller and theory associated with signal processing related to a hybrid SP and DP Heat Pulse Probe. The higher sampling rate allowed for digital filtering to 505 be applied.
• Signal processing was used to determine thermal conductivity using a SP model that did not rely on a late-time SP approximation. A DP model was used to determine changes in the effective DP probe spacing radius.

510
• The DP and SP signal processing models introduced in this paper improved overall estimates of soil water content  w and bulk soil density  for sand and peat soils, indicating that detection of effective changes in the probe spacing radius using signal processing is useful to correct for model error and physical changes in the probe spacing. This improvement is associated with standard HP and DP probes that are used together in a novel fashion along with signal processing. 515 • Further research is required to test the signal processing models introduced in this paper and to compare the estimates of soil water content  w and bulk soil density  to estimates made using other measurement systems and technologies. The effective radius calibration has an advantage for soil types where expansion and contraction of the soil can cause changes in the effective probe spacing radius ( ) rt. 520

Code Availability
The computer code and data used to produce the figures and numerical results in this paper can be downloaded from Figshare

Data Availability
The data from the experiments described in this paper can also be obtained from the Figshare data repository as a separate download (https://www.doi.org/10.6084/m9.figshare.11371455). 530

Appendix A
To efficiently obtain the inverse of ( ) i Ex, endpoints for a search interval need to be appropriately selected, particularly when the inverse model runs on an embedded resource-constrained microcontroller. To ensure numerical continuity and accuracy between the forward and inverse models, the same ( ) i Ex function is used in the inverse of ( ) i Ex in lieu of alternative 535 numerical approximations.
and we need to show: ( ) Using 5.1.20 of Abramowitz and Stegun (1964), we need to show that with 0  x : The RHS can be replaced by a power series representation The proof proceeds by contradiction and shows that the inequality holds. We assume 565 1/ 2 However, this is a contradiction, so inequality (A.1) holds.
From inequality (A.1), the search interval for Ex is computed using Golden-section search (Kiefer, 1953) on the bounded interval. 570 The following shows how endpoints of the search interval for the inverse are selected for the cooling section of the DP curve using the result given above. Without the ( ) /4  qk term, Eq. (4) is of the form:  where data and system operation can be exchanged between machines using commands sent over USB, RS-232, or SDI-12.
The microcontroller acts as a state machine that samples data from the AFE and stores the data in SDRAM where signal processing is conducted. The circuit is powered by a nominal 12V DC supply that is reduced to 3.3V by a DC-DC switcher. 880 The HPP is a hybrid SP and DP design comprised of a heater needle and a sense needle. The effective distance between the 33 needles as a function of time is ( )      Table 2: Heat pulse strengths, time of heating and cooling and total time for each experiment conducted on sand and peat. 1000 The experiment identifier is an alphabetical letter that identifies the experiment set. The # is an indication of the total number of experiments conducted per set.        Table 7: RMSD, MB, and PD comparisons for  .