Calibration and Uncertainty Quantification for Single-Ended Raman-Based Distributed Temperature Sensing: Case Study in a 800 m Deep Coaxial Borehole Heat Exchanger

1. Introduction

Data-driven modeling and learning have become increasingly important in science and engineering. Along with it, the need for reliable measurements with well-characterized uncertainties has grown. The field of geothermal energy is no exception. On the contrary, underground inaccessibility and relatively scarce data make accurate measurements particularly vital to perform correct inference on existing or future systems.

For Ground Source Heat Pump (GSHP) systems in particular, knowledge of the underground temperature field is valuable to understand the underlying heat transfer mechanisms in the ground. In turn, this combined knowledge can be used to improve the performance of the system in several ways, such as enhancing current performance, updating predictions about future operating conditions and providing more information about underground thermo-hydraulic conditions.

One way to obtain information about the underground temperature field is to use Distributed Temperature Sensing (DTS) inside boreholes. DTS enables temperature measurement with some spatial resolution along the sensor itself, a fiber optic cable, rather than in a specific location. Furthermore, these measurements can be performed quasi-simultaneously over the length of the fiber [1,2]. Selker et al. [3] discuss pros and cons of several DTS techniques: fiber Bragg grating, Brillouin and Raman scattering. Most short-distance sensing (<15 km) is based on Raman Optical Time Domain Reflectometry (OTDR) [1].

Raman DTS systems have been used in various fields, including monitoring power transmission lines [4], fire detection and safety [5,6], volcanology [7], oil well monitoring [8], inferring paleoclimate [9,10,11], leakage detection in dams and dikes [12], hydrogeology [13,14], atmospheric science and extraterrestrial geophysics [15].

In shallow boreholes the Raman DTS technique – simply referred to as DTS below – has been used for about three decades with first applications in hydrogeology [2,16,17]. In the reviewed literature, application of the DTS technique to GSHP research first appears in the context of Distributed Thermal Response Testing (DTRT) [18,19,20,21]. DTRTs based on DTS is still a current topic of research and even seems to have regained interest in recent years [22,23,24,25,26,27,28].

Besides their application to DTRTs, fiber optic cables have also been used to monitor operation of Borehole and Aquifer Thermal Energy Storages (BTES and ATES) [29,30,31,32,33]. Additionally, DTS data have indirectly been capitalized on to create thermal conductivity maps [34] or to measure the undisturbed temperature profile and evaluate its effect on design procedures [19,35,36,37].

In-situ DTS measurements is coupled to different challenges, three of which are presented hereafter. The first one, calibration, is described by Selker et al. [3]: "an important aspect to keep in mind is that this is a technology that provides temperature data with minimal setup or interpretation required. It is necessary to calibrate the instrument to the cable by attaching the fiber-optic cable to the instrument with the cable in an environment of well-known temperature”. When performing temperature measurements with a DTS device, in-situ calibration is indeed a requirement [1,38,39,40,41]. While commercial DTS units typically have an in-built calibration procedure, it may not be precise and accurate enough for hydrogeological applications [39,40,42].

Hausner et al. [39] discuss four different calibration methods for single-ended configurations1 and their application to two different case studies. The authors conclude that explicit calculation of calibration parameters leads to the best precision and accuracy. They also admit that this is challenging to achieve for field installations since at least two baths must be maintained at different temperature levels, which is especially demanding for long-term installations [46]. For the best calibration method, Hausner et al. report mean RMSEs of 0.131 K and 0.108 K for a lab and a field case study, respectively. The corresponding mean biases are of 0.016 K and 0.043 K, respectively. These values are calculated over sections of fiber in a validation bath. For the same fiber configuration, the manufacturer’s calibration leads to RMSEs statistics of 0.793 K and 0.583 K for the lab and field case study, respectively, while bias statistics are of 0.792 K and -0.580 K. Notably, the RMSEs are dominated by the bias component for the manufacturer calibration. This suggests that RMSEs might be substantially reduced by removing biases through calibration. Lillo et al. [47] report calibration metrics in a similar way for five different calibration algorithms – including the one from [39] – and five different sites (of which one is a laboratory test) for single-ended duplexed configurations. They find an algorithm that further improves the calibration methods proposed in [39], although "the calibration process do not necessarily fulfill physical considerations". Another important aspect that calibration may help tackle are step losses caused by fusion splices, local strains, fiber damages or tight bends [39,40,43].

The second challenge that may arise with in-situ DTS measurements are temperature drifts caused by changing ambient conditions [3,39,40,41,43]. In particular, changing ambient temperature may lead to some disturbances in the internal oven of the DTS instrument used for the manufacturer’s calibration. Hence, the need for dynamic calibration is highlighted in the literature [46].

The third challenge with DTS measurements is the quantification of uncertainty [37], which is more complex in field setups. Tyler et al. [40] study spatial and temporal repeatability, as well as spatial resolution2, for three different instruments. For the temporal repeatability, the authors find standard deviations of 0.08, 0.13 and 0.31 K for biases of 0.33, 0.18 and 0.14 K for each instrument respectively. For the spatial repeatability, standard deviations are of 0.02, 0.04 and 0.08 K and the biases are not given although it appears graphically that accuracy is better for the instruments with larger standard deviations. The authors report the use of a single calibration bath that was short (≈ 5 m) due to logistical challenges. They also characterize their temporal repeatability test as "fairly short" (2h). For the spatial resolution, average values between 1.91 to 3.45 m are reported.

Des Tombes et al. [45] propose an new calibration approach allowing for the quantification of uncertainty for the temperature. The temperature uncertainty solely accounts for the noise from the detectors and the uncertainty in the estimated parameters. They find a nearly constant 95% confidence bound of about ±0.3 K for an experiment with a 100 m fiber with a double-ended setup and three baths. Among other things, they observe that the contribution from the parameter estimation is small compared to the noise of the Stokes and anti-Stokes.

When DTS is applied for GSHP applications, it is in most cases unclear if calibration baths are used and how this affects the measurement uncertainty. In most cases, precision (a standard deviation value) is reported [18,25,33,38,50]. The reported precisions are in general in the order of 0.01–0.1 K which is consistent with what is otherwise presented [3,40]. Accuracy is more seldom discussed. Monzó [51] finds mean bias as high as 1.86 K for a 2.4 km long duplexed installation with six splices along the fiber length (not including the far-end splice). For the forward part of the duplexed configuration, the author finds a maximum systematic error of -0.51 K. Abuasbeh and Acuña [29] report a maximum and mean systematic error of -0.19 K and 0.09 K, respectively, in a validation bath. Acuña et al. [18] theoretically evaluate the systematic error that could arise from the undetermined position of the fiber inside the Borehole Heat Exchanger (BHE) under laminar flow conditions. They find a maximum possible bias of ±0.4 K for this effect. Fujii et al. [21] use the ±1.0 K accuracy and 0.1 K resolution specified by the manufacturer.

For the GSHP field, there is a lack of consistent metrics for characterization of uncertainty. Moreover, there are some specific issues with DTS measurements. For instance, one should keep in mind that the obtained temperatures are averages in time and space. As such it seems needed to specify which sampling interval and temporal averaging are used when providing metrological metrics. Given a long enough integration time under repeatability conditions [49], one can get any arbitrarily low random error for the measured average temperatures. For a real case, however, this metric might not be the most relevant, especially if the dynamics are important. In such a case, the uncertainty of the underlying temperature – rather than its average, might be more relevant.

In the overall reviewed literature, metrological metrics are often presented for calibration baths3, validation baths or specific sections of fiber but it is not always clear how these metrics should be propagated to the rest of the fiber optic cable. Additionally, it is not always possible to perform an optimal calibration – i.e. a dynamic calibration with at least three different calibrated sections as suggested in [39] – e.g. for logistics constraints [37,40,46].

In this paper, a new calibration method for single-ended DTS configurations in a BHE with a single bath is proposed. In addition, a method for removing fictitious temperature drifts due to variations in ambient temperature is presented. Finally, and more importantly, the DTS measurement uncertainty is quantified for a DTRT in a 800 m deep borehole with coaxial BHE. Among other things, the uncertainty includes components from the Stokes/anti-Stokes signals, the calibration bath temperature and the parameter estimation used during calibration.

2. Materials and Methods

2.1. Measurement principle

In most cases, when light is scattered by molecules, the scattering will be elastic so-called Rayleigh scattering. In other words, the scattered light will have the same frequency than the incident light. Sometimes, however, the scattered light will have a different frequency. When light is scattered by a molecule, the incident photon energy may turn into a phonon (a molecular quantum vibrational excitation). The scattered light would then have a lower frequency (lower energy) than that of the incoming light. This phenomenon is referred to as Stokes Raman scattering. On the other hand, the energy of a phonon may be captured by a scattered photon, meaning that scattered light would have higher frequency. This is referred to as anti-Stokes. According to Bose-Einstein statistics, the ratio of Stokes/anti-Stokes intensities is proportional to [52,53,54,55]

$$\frac{P_S}{P_{AS}}\propto e^{\frac{\Delta E}{k_{B}T}},$$

(1)

where S and $AS$ refer to Stokes and anti-Stokes, $\Delta E=\hslash \Delta \omega$ is the difference in energy related to the frequency shift between incident and scattered light, which also correspond to the difference in the molecules vibrational energy levels (phonon), $k_B$ is the Boltzmann constant and T is the temperature. From eq.1, it is clear that the Stokes/anti-Stokes signals may be used to infer local temperatures. Accounting for the fraction of the scattered light that reaches back to the instrument, the signal attenuation within the fiber (Beer’s law), the efficiency of the detector and potential step losses, the temperature may be expressed as [39,55]

$$T(z,t)=\frac{\gamma }{ln\left(\frac{P_S(z,t)}{P_{AS}(z,t)}\right)+C\left(t\right)-\Delta R\left(z\right)-\Delta \alpha z},$$

(2)

where z is the distance from the instrument (inside the fiber), t is time, $\gamma =\frac{\Delta E}{k_B}$, C accounts for the fraction of light that is back-scattered to the instrument and the instrument detection efficiency, $\Delta R\left(z\right)={\sum }_{n=1}^N\Delta R_n\theta (z-z_n)$ represents the potential (N) step losses and $\Delta \alpha$ is the difference in the power attenuation coefficients for the Stokes and anti-Stokes signals. For more in-depth discussions about the Raman DTS measurement principle the reader is referred to [52,53,55] and the supplementary material of [39].

2.2. In-situ DTS configuration

The experimental setup for this study is presented in Figure 1. It consists of a Sensornet Halo-DTS^TM unit connected to two ca. 1050 m long multimode optical fibers (Solifos BRUsens temperature sensing cables) in a single-ended configuration. The fibers are inserted into a 800 m deep water-filled borehole equipped with a coaxial BHE (see [56] for more details on the BHE). The temperature measurements are performed during a heating DTRT. In this paper, distances inside the borehole may be referred to as borehole depth or length although there might be a difference between the two (if the borehole is unknowingly inclined for instance). The borehole length should not be confused with the actual distance along the fiber cable. The fiber cables are armored with stainless steel and placed in a Nylon sheath to hold the water pressure. A fusion splice between connector patch cords and the fiber channels exist about 5 meters fiber length from the DTS unit. The DTS unit has a 50 m long internal fiber coil used for dynamic adjustment of the parameter $C\left(t\right)$ in eq.2. This is done using a reference temperature sensor and a reference fiber section (-34.4 to -6.0 m) placed in a internal oven, i.e. in which air temperature is supposedly uniform. The internal fiber coil has a different differential loss term compared to the external fiber cables. This must be accounted for in eq.2. Moreover, an assumed step loss is applied at the fiber-instrument connection (connector) and the step loss at the splice must be accounted for such that

$$T(z,t)=\left\{\begin{array}{cc}\frac{\gamma }{ln\left(\frac{P_S(z,t)}{P_{AS}(z,t)}\right)+C\left(t\right)-\Delta {\alpha }_{int}(z+d_{int})}\hfill & \mathrm{for}z<0.\hfill \\ \frac{\gamma }{ln\left(\frac{P_S(z,t)}{P_{AS}(z,t)}\right)+C\left(t\right)-\Delta R_{0}u\left(z\right)-\Delta R_{5}u(z-5)-\Delta \alpha z-\Delta {\alpha }_{int}{d}_{int}}\hfill & \mathrm{for}z\ge 0.\hfill \end{array}\right.,$$

(3)

where $\Delta {\alpha }_{int}$ and $d_{int}$ are the internal differential losses and coil length, respectively. Note that the connector is used as reference for the distance ($z=0$). The parameters $C\left(t\right)$, $\Delta R_0$ and $\Delta {\alpha }_{int}$ can be found by reverse-engineering.

A calibration bath is placed right before the fiber cables go into the borehole. The bath is initially filled with a water and ice and well-mixed in order to have a uniform temperature of 0°C in the bath. For both fiber cables, the length inside the calibration bath is about 46.7 m (between fiber distances of 194.8 m and 241.5 m for the first cable and between 207.0 and 253.7 m for the second cable). Figure 2 shows the Stokes and anti-Stokes lines for both fiber cables at about 3.3 hours after the data acquisition start. Measurements are performed with an integration time of 5 minutes and a sampling interval of 2 m. The different sections (internal oven, patch cord, ambient air, borehole) and remarkable points (connector and fusion splice) are also shown in the figure.

Note that only one calibration bath is used. This is sub-optimal for calibration since there are usually three parameters to determine, so one would need at least three reference sections (three baths) [39]. In particular, having only one calibration bath may be challenging for the determination of the differential loss term $\Delta \alpha$. For DTS application in boreholes, it is clearly not possible to place a calibration bath at the end of the fiber cable. Since there are four fiber channels in each fiber cable, one could have spliced the fiber onto itself, a so-called single-ended duplexed configuration. While this should be done whenever possible, it requires a special termination unit that can both hold the pressure and is small enough to fit in the BHE, which is not always practically or economically feasible. In addition, two extra splices would be introduced at the bottom which would add some complexity to the calibration procedure.

2.3. Correction of temperature drifts

As discussed in the introduction, changing ambient conditions may lead to fictitious temperature drifts in the DTS measurement [3,39,40,41,43]. Some units can manage larger variations of ambient conditions but this may still remain an issue. Dynamic calibration may be performed with the right configuration, although this is not always possible.

Here, the DTS unit is placed inside a container in which the conditions are not controlled. As a result, raw temperature data show a daily oscillation as seen in Figure 3. The figure shows the evolution of temperature at two different sections within the borehole (around 200 and 700 m) during the different phases of the DTRT: undisturbed ground, circulation, heat injection and heat recovery. It is quite clear that there is an oscillating pattern throughout time. Now, the question is: is this oscillation real or is it fictitiously induced by the measuring device? During the heat injection phase of the DTRT - between about 20 to 230 hours - it is unlikely that these fluctuations are real, but it cannot be completely excluded since the water being circulated inside the BHE could be influenced by above-ground ambient conditions (e.g. these osciallations are also noticed in the sections of the fiber located outside the borehole).

During the heat recovery phase, however - between about 230 hours until the end of the test - it is highly unlikely that these fluctuations are real because the water circulation is stopped, so there is no reason to think that the ambient conditions above ground would affect the temperature 200 m, let alone 700 m, below ground. This can be therefore be used as a condition for removing these oscillations. Since $\gamma$ is constant, and since the step losses and differential losses are considered time-independent, the potential correlation can be determined to a good approximation using the inverse of the temperature (see Appendix A). Hence, the correction may be expressed as $\beta \left(T_{ref}\left(t\right)-{\overline{T}}_{ref}\right)$ where $\beta$ is the regression coefficient such that

$$\beta \left(T_{ref}\left(t\right)-{\overline{T}}_{ref}\right)=-\frac{1}{N_{{\mathcal{Z}}_{bh}}}\sum _{z\in {\mathcal{Z}}_{bh}}ln\left(\frac{P_S(z,t)}{P_{AS}(z,t)}\right)-C\left(t\right),t\in {\mathcal{T}}_{rec}.$$

(4)

This correction is added to the denominator of eq.2. Here, ${\mathcal{Z}}_{bh}$ is taken as the whole borehole starting from 15 m below the surface to make sure that any influences from above ground temperature are excluded. $N_{{\mathcal{Z}}_{bh}}$ is the number of measurement sections within ${\mathcal{Z}}_{bh}$ while the time span ${\mathcal{T}}_{rec}$ is taken as the recovery period, between 261 and 355 hours, since the water circulation is stopped during that period. Undisturbed conditions can also be used, given a long enough period of measurements (e.g. if the recovery period is missing).s

2.4. Calibration

As noted above, there is only one calibration section for each fiber. This is a sub-optimal configuration for determining the unknown parameters in eq.2, i.e. $\gamma$, $C\left(t\right)$, $\Delta R\left(z\right)$ and $\Delta \alpha$. The first term, $\gamma$, is mostly dependent on the DTS instrument. Thus, it is reasonable to use the default value, thereby having one less parameter to estimate. In addition, $C\left(t\right)$ is already calibrated by the instrument and corrected according to the previous section. Hence, this parameter may also be removed from the parameter estimation list.

Hence, only $\Delta R\left(z\right)$ and $\Delta \alpha$ remains to be estimated. Here the temperature between the connector and the fusion splice (0-5 m) is not of interest so it is sufficient to use the default step loss value at the connector and calibrate the step loss at 5 m. Another way to state this is that the step losses are aggregated into one value only with the consequences that temperatures between 0 and 5 m of fiber cable may not be inaccurate. Accordingly, the parameter estimation problem turns into a simple linear regression with the fiber length as a variable

$$\Delta \alpha ·z+\Delta R=\frac{1}{N_{{\mathcal{T}}_{bath}}}\sum _{t\in {\mathcal{T}}_{bath}}\left(ln\left(\frac{P_S(z,t)}{P_{AS}(z,t)}\right)+C\left(t\right)\right)-\frac{\gamma }{T_{bath}}-\Delta {\alpha }_{int}{d}_{int},z\in {\mathcal{Z}}_{bath}.$$

(5)

The bath sections used for calibration, ${\mathcal{Z}}_{bath}$, are taken as 198.9-237.5 m for the first fiber cable and 211.0-249.6 m for the second cable. That is, they are taken slightly shorter than the total bath section to avoid any temperature bias due to spatial resolution where the fiber enters and exits the bath [48]. The temperature of the bath, $T_{bath}$, is maintained at 0°C between 0.5 to 6.85 hours after start of data acquisition (${\mathcal{T}}_{bath}$). Potential variations in this temperature are accounted for in the uncertainty analysis.

The calibration is performed for each fiber cable separately although - as noted in previous sections - it is expected that this will give a poor estimation of $\Delta \alpha$ since the linear regression of eq.5 is performed over a limited section of fiber. Another approach is to combine the calibration of the two fiber cables by stating that the bottom temperature should be equal for both the cables at all times. This leads to the following equation

$$(\Delta {\alpha }_1-\Delta {\alpha }_2)z+\Delta R_1-\Delta R_2=\frac{1}{N_{\mathcal{T}}}\sum _{t\in \mathcal{T}}\left(ln\left(\frac{P_{S,1}(z,t)·P_{AS,2}(z,t)}{P_{AS,1}(z,t)·P_{S,2}(z,t)}\right)+C_1\left(t\right)-C_2\left(t\right)\right),z\in {\mathcal{Z}}_{bottom},$$

(6)

where the subscripts 1 and 2 refer to the first and second fiber cables (or channels). The bottom section, ${\mathcal{Z}}_{bottom}$, could just be taken as the bottom-most measurement section although that risks giving too much importance to measurement noise at this length. Instead, a slightly longer section may be considered if the temperatures in the two legs of the BHE are known not to vary significantly (as in the bottom of a deep borehole for instance). Note that $\mathcal{T}$ represents the whole test period. Potential deviations from temperature equality at the borehole bottom are investigated in the uncertainty analysis.

The previous expressions may be combined in a compact block matrix form

$$\left[\begin{array}{cc}{\mathbf{X}}_{\mathbf{1}}& \mathbf{0}\\ \mathbf{0}& {\mathbf{X}}_{\mathbf{2}}\\ {\mathbf{X}}_{{\mathbf{12}}_{\mathbf{1}}}& -{\mathbf{X}}_{{\mathbf{12}}_{\mathbf{2}}}\end{array}\right]\left[\begin{array}{c}\Delta R_1\\ \Delta {\alpha }_1\\ \Delta R_2\\ \Delta {\alpha }_2\end{array}\right]=\left[\begin{array}{c}{\mathbf{Y}}_{\mathbf{1}}\\ {\mathbf{Y}}_{\mathbf{2}}\\ {\mathbf{Y}}_{\mathbf{12}}\end{array}\right],$$

(7)

where, for $p\in \left\{1,2\right\}$

$${\mathbf{X}}_{\mathbf{p}}=\left[\begin{array}{cc}1& z_{p,1}\\ ⋮& ⋮\\ 1& z_{p,n}\end{array}\right]{\mathbf{Y}}_{\mathbf{p}}=\left[\begin{array}{c}{y}_{p,1}\\ ⋮\\ y_{p,n}\end{array}\right]{\mathbf{X}}_{{\mathbf{12}}_{\mathbf{p}}}=\left[\begin{array}{cc}1& z_{p,m-q}\\ ⋮& ⋮\\ 1& z_{p,m}\end{array}\right]{\mathbf{Y}}_{\mathbf{12}}=\left[\begin{array}{c}{y}_{12,m-q}\\ ⋮\\ y_{12,m}\end{array}\right]$$

(8)

The integer n corresponds to the n-th section in the calibration bath, while m corresponds to the section at the borehole bottom and q indicates the number of extra sections taken into the equality condition expressed in eq.6. For instance, if q equals 0, only the bottom measurement section is considered while if q equals 1, the equality condition is applied to both the last and penultimate sections.

For $p\in \left\{1,2\right\}$ and $l\in \left[1,n\right]$, the terms $y_{p,l,k}$ of the vector ${\mathbf{Y}}_{\mathbf{p}}$ may be expressed as the right-hand side of eq.5

$$y_{p,l}=-\frac{\gamma }{T_{bath}}+\overline{ln\left(\frac{P_{Sp,l}}{P_{ASp,l}}\right)}-\Delta {\alpha }_{int}{d}_{int}+\overline{C_p}.$$

(9)

Similarly, for $l\in \left[0,q\right]$, the terms $y_{12,m-l,k}$ of the vector ${\mathbf{Y}}_{\mathbf{12}}$ may be expressed as the right-hand side of eq.6

$$y_{12,m-l}=y_{1,m-l}-y_{2,m-l}=\overline{ln\left(\frac{P_{S1,m-l}}{P_{AS1,m-l}}\frac{P_{AS2,m-l}}{P_{S2,m-l}}\right)}+\overline{C_1}-\overline{C_2},$$

(10)

where the averages in the two previous equations are with respect to time.

2.5. Uncertainty evaluation

2.5.1. Propagation of error

Once the calibration performed, a natural question that arise is that of the uncertainty. The temperature uncertainty is determined using the state-of-the-art definitions, methods and procedures described in [49], in particular regarding the propagation of uncertainty (combined uncertainty). The temperature measurement can be modeled by eq.3 and the linear regression for the calibration expressed in eqs.5,6,7. Thus, the relative temperature uncertainty may be expressed as

$$\begin{array}{cc}\hfill \frac{u_c\left(T\right)}{T}=\frac{T}{\gamma }(& u^2\left(ln\left(\frac{P_S}{P_{AS}}\right)\right)+u^2\left(C\right)+\left(T_{ref}-{\overline{T}}_{ref}\right)u^2\left(\beta \right)+u^2(\Delta R_{lin.reg.})\hfill \\ \hfill & +z^2·u^2(\Delta {\alpha }_{lin.reg.})+2z·u(\Delta R_{lin.reg.},\Delta {\alpha }_{lin.reg.}){)}^{1/2}.\hfill \end{array}$$

(11)

Note that the combined uncertainty, $u_c\left(T\right)$ depends on both the distance and time - as does the temperature. The term $u\left(X\right)$ stands for standard uncertainty of quantity X.

The standard uncertainty related to the Stokes and anti-Stokes is determined using a condition when the underlying temperature is expected to be constant. Then, all the observed variations in temperatures are attributed to variations in the measured Stokes and anti-Stokes signals such that

$$u\left(ln\left(\frac{P_S}{P_{AS}}\right)\right)=u\left(T_{DTS}\right)\frac{\gamma }{T_{DTS}^2}$$

(12)

There may be other effects affecting the temperature reading during the chosen period (e.g. variation in the internal oven temperature and its influence on C or actual variations due to fluid movement inside the BHE) but these effects are considered small since the period is short. Moreover, even if such effects are present, the main consequence would be that the uncertainty attributed to the Stokes and anti-Stokes readings is slightly overestimated, hence leading to a somewhat conservative evaluation of uncertainty. It must be clear that the uncertainty of eq.12 is the uncertainty of the Stokes and anti-Stokes averaged over the chosen integration period (here five minutes) and not the uncertainty of every single Stokes and anti-Stokes signal that is measured by the instrument within that period (before internal processing). Note that the temperature directly obtained from the instrument, $T_{DTS}$, is used. This is due to avoid introducing any other type of error in the standard uncertainty.

Using eq.3 ($z<0$), the internal oven and its reference temperature, one can compute different values for the parameter $C\left(t\right)$ over the whole internal reference section. The standard uncertainty of the parameter C is then simply taken as the standard deviation of the different C samples within the internal reference section.

The standard uncertainty of the correction term may be computed from the standard deviation of the residuals, ${\widehat{\sigma }}_{\epsilon }$, as

$$u\left(\beta \right)=\frac{{\widehat{\sigma }}_{\epsilon }}{\sqrt{{\sum }_{t\in {\mathcal{T}}_{rec}}{\left(T_{ref}\left(t\right)-{\overline{T}}_{ref}\right)}^2}}=\sqrt{\frac{1}{(N_{{\mathcal{T}}_{rec}}-1)}\frac{{\sum }_{t\in {\mathcal{T}}_{rec}}{\widehat{\epsilon }}^2\left(t\right)}{{\sum }_{t\in {\mathcal{T}}_{rec}}{\left(T_{ref}\left(t\right)-{\overline{T}}_{ref}\right)}^2}}.$$

(13)

This assumes that the residuals are i.i.d. The uncertainty components related to the linear regression coefficients may be estimated in a similar way, i.e through the covariance matrix, $\Sigma$, of coefficient estimates

$$\Sigma ={\widehat{\sigma }}_{\epsilon }^2{\left({\mathbf{X}}^T\mathbf{X}\right)}^{-1},$$

(14)

where the matrix $\mathbf{X}$ is the leftmost block matrix of eq.7. The variance terms of the linear regression coefficients are on the diagonal while the rest of the matrix terms are covariances. Note that ${\widehat{\sigma }}_{\epsilon }$ is a generic term for the standard deviation estimate of the residuals but it is not the same in eq.13 and eq.14.

2.5.2. Deviations in bath and bottom conditions

Up to this point, it has been assumed that the bath temperature was fixed at 0°C and that there is a perfect temperature equality between the two fiber cables at the bottom of the borehole. While these are most reasonable assumptions to make, it is not inconceivable that small deviations in those conditions might exist. How would such small deviations impact the calibration and the temperature uncertainty? This question cannot be answered simply using the propagation of error method. Thus, a Monte-Carlo study is performed in order to answer that question. Two random variables are used for the study such that

$$\left\{\begin{array}{c}{T}_{bath}^{\prime }\sim \mathcal{N}(T_{bath},{\sigma }_{bath}^2)\hfill \\ T_2(z,t)=\psi T_1(z,t)\forall (z,t)\in ({\mathcal{Z}}_{bottom},\mathcal{T}),with\psi \sim \mathcal{N}(1,{\sigma }_{bottom}^2)\hfill \end{array}\right..$$

(15)

Note that the random variables are consistent in time, meaning that they behave like systematic errors and not re-sampled at every time step. The temperature $T_{bath}^{\prime }$ is the the actual bath temperature, while $T_{bath}$ is the temperature that is most reasonable to assume in the absence of any other information. Here, the latter is taken as 0°C (273.15 K) while the standard deviation ${\sigma }_{bath}$ is assumed to be 0.1 K. $T_1$ and $T_2$ refer to temperatures from the first and second channel, respectively. The standard deviation of the ratio is assumed as 0.05%. As an example, a $T_1$ equal to 20°C (293.15 K) leads to a standard deviation of the difference of about ± 0.15 K. The potential deviation at the bottom is written as ratio rather than a additional correction so that the linear regression of eq.7 may still be applied with a small modification

$$\left[\begin{array}{cc}{\mathbf{X}}_{\mathbf{1}}& 0\\ 0& {\mathbf{X}}_{\mathbf{2}}\\ {\mathbf{X}}_{{\mathbf{12}}_{\mathbf{1}}}& -{\mathbf{X}}_{{\mathbf{12}}_{\mathbf{2}}}\end{array}\right]\left[\begin{array}{c}\Delta R_1\\ \Delta {\alpha }_1\\ \psi \Delta R_2\\ \psi \Delta {\alpha }_2\end{array}\right]=\left[\begin{array}{c}{\mathbf{Y}}_{\mathbf{1}}\\ \psi {\mathbf{Y}}_{\mathbf{2}}\\ {\mathbf{Y}}_{\mathbf{12}}\left(\psi \right)\end{array}\right],$$

(16)

where the terms $y_{12,m-l}$ of vector ${\mathbf{Y}}_{\mathbf{12}}\left(\psi \right)$ are expressed as

$$y_{12,m-l}=\overline{ln\left(\frac{P_{S1,m-l}}{P_{AS1,m-l}}\right)}+\overline{C_1}-\psi \left(\overline{ln\left(\frac{P_{S2,m-l}}{P_{AS2,m-l}}\right)}+\overline{C_2}\right).$$

(17)

The estimated parameters for the second fiber may then be corrected by $1/\psi$ in order to find the parameters of interest. The Monte-Carlo study is performed for the bath condition only, for the bottom condition only and for a combination of both. This is done to identify the contribution of each condition. Thus, the estimated parameters may be decomposed into different terms such that

$$\left\{\begin{array}{c}\Delta R=\Delta R_{lin.reg.}+\delta R_{bottom}+\delta R_{bath}\hfill \\ \Delta \alpha =\Delta {\alpha }_{lin.reg.}+\delta {\alpha }_{bottom}\hfill \end{array}\right..$$

(18)

where $\delta$ symbolized random components resulting from the Monte-Carlo study. Note that the bath condition only affects the intercept terms in the linear regression of eqs.5 and 7 since it just acts as an offset for the y’s terms. In terms of the uncertainty, this leads to a six extra variance and covariance terms

$$\begin{array}{cc}\hfill \frac{u_c\left(T\right)}{T}=\frac{T}{\gamma }(& u^2\left(ln\left(\frac{P_S}{P_{AS}}\right)\right)+u^2\left(C\right)+\left(T_{ref}-{\overline{T}}_{ref}\right)u^2\left(\beta \right)\hfill \\ \hfill & +u^2\left(\delta R_{bottom}\right)+u^2\left(\delta R_{bath}\right)+2u(\delta R_{bottom},\delta R_{bath})\hfill \\ \hfill & +z^2{u}^2\left(\delta {\alpha }_{bottom}\right)+2z·u(\delta R_{bottom},\delta {\alpha }_{bottom})+2z·u(\delta R_{bath},\delta {\alpha }_{bottom})\hfill \\ \hfill & +z^2·u^2(\Delta {\alpha }_{lin.reg.})+u^2(\Delta R_{lin.reg.})+2z·u(\Delta R_{lin.reg.},\Delta {\alpha }_{lin.reg.}){)}^{1/2}.\hfill \end{array}$$

(19)

2.5.3. Temperature averaging

So far, the uncertainty has been expressed for the temperature that is averaged during a user-chosen integrating time. As discussed in the introduction, the uncertainty of the average temperature becomes arbitrarily low for a long enough integration time. The underlying true temperature will, however, most likely change during the integrating time. If the underlying temperature is the actual measurand of interest, the uncertainty related to the temperature dynamics under the integration period should therefore be taken into account. The temperature at any time $t+\Delta t$ between two integrating times t and $t+\Delta t_{int}$ may be decomposed into

$$T(z,t+\Delta t)={\overline{T}}_{\Delta t_{int}}\left(z\right)+\Delta T(z,t+\Delta t).$$

(20)

The temperature dynamics obviously depends on the application and conditions specific to each measurement. It is therefore hard to say something general about the error term $\Delta T(z,t+\Delta t)$. Nevertheless, assuming that the integrating time is short enough so that the temperature may be approximated as a piece-wise linear function in the time domain, the error term $\Delta T(z,t+\Delta t)$ may be expressed as

$$\Delta T(z,t+\Delta t)=\frac{\partial T}{\partial t}(z,t)(\Delta t-\frac{\Delta t_{int}}{2}).$$

(21)

Thus the error at a given time and distance under such conditions may be treated as a random variable such that

$$\Delta T\sim U\left(-\left|\frac{\partial T}{\partial t}\right|\frac{\Delta t_{int}}{2},\frac{\partial T}{\partial t}\frac{\Delta t_{int}}{2}\right).$$

(22)

In that case, the uncertainty related to averaging may be expressed as

$$u(\Delta T)=\left|\frac{\partial T}{\partial t}\right|\frac{\Delta t_{int}}{\sqrt{12}}=\frac{\left|\Delta T_{max}\right|}{\sqrt{12}}.$$

(23)

Not surprisingly, the standard uncertainty related to averaging is directly proportional to the first-order derivative (or maximum temperature change). For TRTs, this mean that the uncertainty will be largest when the heat injection start or stop and will quickly decrease after. According to data in [56] and [57], the standard uncertainty is always below 0.1 K after the first hour of heat injection.

2.5.4. Overall uncertainty, degrees of freedom and extended uncertainty

The standard uncertainty related to averaging and the standard uncertainty of eq.19 may be combined to compute the overall standard uncertainty such that

$$u_c\left(T\right)=\sqrt{u_c^2\left(\overline{T}\right)+u^2(\Delta T)}.$$

(24)

This standard uncertainty may then be extended to any given confidence level as

$$U\left(T\right)=t_{1-\frac{\zeta }{2},{\nu }_{eff}}{u}_c\left(T\right),$$

(25)

where $t_{1-\frac{\zeta }{2},{\nu }_{eff}}$ is the $1-\frac{\zeta }{2}$ quantile of the Student t distribution with ${\nu }_{eff}$, the effective degrees of freedom. The effective degrees of freedom is calculate from the Welch-Satterthwaite formula as reported in [49].

Figure 4. Temperature time profiles after calibration and correction of temperature drifts at four different borehole depths: (a) 50 m, (b) 150 m, (c) 400 m and (d) 750 m.

Figure 4. Temperature time profiles after calibration and correction of temperature drifts at four different borehole depths: (a) 50 m, (b) 150 m, (c) 400 m and (d) 750 m.

3. Results

3.1. Correction of temperature drifts and calibration

Temperatures resulting from the calibration - including the correction of temperature drifts - are shown in Figure 3. The figure shows temperature profiles in time at different depths and for the two fiber channels. Besides the raw data, the results from three calibration methods are shown: on-site calibration using the instrument software, segregated calibration (i.e. each channel is calibrated on its own without the bottom temperature equality condition of eq.6) and combined calibration. Segregated and combined calibration methods also include the correction of temperature drifts.

The raw data and on-site calibration show daily oscillations that are typical of temperature drifts in the internal oven. As mentioned earlier, it is unlikely that such daily variations are real in the borehole, especially during the recovery phase when circulation is turned off.

Another issue is the inequality in temperature between the two channels during the recovery period. This happens for the raw data, the on-site calibration and the segregated calibration, for which the temperature inequality is particularly large.

The combined calibration remedies to both of these issues. It also leads to matching temperatures at the bottom of the borehole - by definition - which can be seen in Figure 5. It is also an encouraging indication that the undisturbed temperature is very similar for both channels and that the temperatures at the top of the borehole are nearly equal during circulation.

The parameters obtained through the different calibration procedures are given in Table 1 while Table 2 gives the fixed parameters during calibration, including the temperature drift correction $\beta$. The default differential loss and those found for the combined calibration are similar. However, not accounting for the step loss due the fusion splice leads to a significant difference in temperatures as seen in Figure 5. It is noteworthy that the temperature profiles from the raw data match at the bottom of the borehole, without any prior constraints unlike the combined calibration procedure. On the other hand, the segregated calibration does not lead to satisfactory results at the bottom of the borehole. This is mostly due to the large differences in differential loss terms, $\Delta \alpha$, which themselves arise from the linear regressions over a small section (calibration bath) compared to the whole fiber length. The on-site calibration leads to a somewhat better temperature match at the bottom. In this case, it would been better to solely calibrate the step loss, $\Delta R$, instead of calibrating both the differential and step losses.

3.1.1. Verification of the calibration

To completely validate the proposed calibration procedure, an independent measurement would be required. This is unfortunately not available for this site. Nevertheless, another DTS measurement in the same borehole was performed about one year prior to the measurements previously described. The same instrument and fiber cables are used, although the channels are switched. Applying the same calibration procedure, the temperature profiles shown in Figure 6 are obtained.

The RMSE over the whole borehole length (top 15 m excluded to avoid influences from air temperatures) are of about 0.18 and 0.16°C for the first and second fiber, respectively (i.e. the first and second channels as presented in this paper). In addition, the values found for the differential loss term are very similar for the first and second fiber cable, which is expected since the two cables are the same product.

From both the visual comparison in Figure 6 and the numerical RMSE indicators, the calibration procedure appears to be consistent for the two different measurements. The raw data show for instance a much weaker consistency as may be seen in Figure 6.

Nevertheless, the estimated parameters appear to be different from one year to another as shown in Table 1. For $\Delta R$, this is expected since disconnecting and re-connecting fiber cables to the DTS instrument will lead to different step losses. For the differential losses $\Delta \alpha$, however, the difference is more surprising because this parameter is supposedly only dependent on the fiber cable itself. In the next section, the uncertainty bounds for $\Delta \alpha$ will help clarify this issue.

3.2. Uncertainty evaluation

Using the methodology developed in Section 2.5, the temperature uncertainty may be computed for all times and distances. The uncertainty in time appears to be relatively constant as can be seen in Figure 7, which shows the calibrated temperature profiles vs. time at the same four depths as shown in Figure 4. The figure hints at an increasing temperature uncertainty with increasing distance. This is confirmed by Figure 8 where vertical temperature profiles in the borehole are shown together with their 95% confidence bounds at three different times. Here again, the chosen times are the same as in Figure 5.

In average, the confidence bounds reach a minimum at around 10 m depth, probably due to influence from the top boundary effects that are reflected in the Stokes/anti-Stokes uncertainty component at the top of the borehole. Except for the topmost 10 m, the 95% confidence bounds show an increasing, non-linear trend as the distance increases (or borehole length). For channel 1, the expanded uncertainty increases from around 0.35 K at 10 m to about 1.65 K at the bottom, while it is from around 0.4 K to 1.65 K for channel 2. At around 300 m, the expanded uncertainty is about 0.74 K for both channels. Values of the expanded uncertainty for different depths are shown in Table 34.

The fact that the two expanded uncertainty are very similar can be seen as a sign of robustness. The manufacturer’s temperature resolution [58] is also given in Table 3 although the lack of clarity around the term makes it hard to compare with the uncertainty figures. According to [45], the temperature resolution can be understood as a standard uncertainty, in which case the uncertainty bounds found in this study are about one degree of magnitude higher than what is reported in product data sheets.

The contribution from the different components of the combined uncertainty is shown in Figure 9 for both channels. Note that the figure show the combined variance instead of the combined standard uncertainty. This is because the different uncertainty components add up in quadrature instead of a simple addition. Thus, simply stapling the standard uncertainty of each component would not be indicative of their contribution to the combined uncertainty. The combined variance and its different component are shown for a given time (300 hours) but they are close to constant in time so the figure is representative for other moments.

The linear regression component clearly dominates the combined variance for larger depths while it almost vanishes close to the borehole top. This is expected: close to the calibration bath, the calibration uncertainty will be lower and vice-versa. For the same reason, it makes sense that the variance related to potential systematic errors in the bath temperature also increases with the distance, although it is not noticeable in the figures. Similarly for potential systematic errors in the bottom equality condition, which contribution is small but can be seen for a small section close to the borehole bottom in the zoomed areas at the top of the two sub-figures. The lower zoomed areas show that the contribution is vanishingly small at shallow depths. Somewhat unexpectedly, the contribution from the depth-constant $C\left(t\right)$ show a depth-dependency. The variance $u^2\left(C\right)$ is constant for a given time – as expected – but the latter is multiplied by $\frac{T^4}{{\gamma }^2}$ (see eq.19) to obtain the contribution to the temperature combined variance, which creates the depth-dependency. The contribution from both the correction based on the internal temperature and averaging is negligibly small. The former is not even visible on any of the zoomed areas of Figure 9. The latter is even negligible during periods in which temperature changes are higher - e.g. at the beginning of the heat injection.

With regard to the previous discrepancy found between the differential loss term during the two calibrations mentioned in the previous section: the found 95% confidence bounds for the differential loss of channel 1 are $9.4739\pm 1.172·{10}^{-5}{m}^{-1}(\nu =40)$ and $8.4853\pm 1.160·{10}^{-5}{m}^{-1}(\nu =25)$, while they are $9.4630\pm 1.190·{10}^{-5}{m}^{-1}(\nu =40)$ and $8.6884\pm 1.146·{10}^{-5}{m}^{-1}(\nu =25)$ for channel 2. Hence, the values found during the two different calibrations are compatible with one another. The difference may be due to a shorter calibration section in the independent calibration. Note that covariance with the step losses is not considered since the latter effectively change from one calibration to the other - due to disconnection and re-connection of the fiber cables.

4. Discussion

DTS offers interesting perspectives in BHE and GSHP applications and has gained interest in recent years, be it for monitoring, field tests (DTRTs), validation of heat transfer models, estimation of the geothermal gradient, evaluation of undisturbed temperature or thermal conductivity maps [19,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,47].

In this study, a new calibration method and quantification of uncertainty for single-ended Raman-based DTS are proposed, as well as a correction for fictitious temperature drifts caused indirectly by ambient temperature variations. The method is developed to address sub-optimal calibration setups since those are fairly common in BHE applications. For instance, it is not always possible to use the minimum recommended of two calibration baths [39] and field conditions may lead to systematic errors (e.g. due to ambient temperature variations). In addition, the far-end of the fiber cable is not usually accessible (since the fiber cable is inside the borehole and not spliced at the bottom). This makes the calibration of the differential loss term $\Delta \alpha$ more challenging. The proposed calibration method is based on one calibration bath only and uses temperature equality between the two fiber cables as an extra constraint. In addition, a correction for fictitious temperature oscillation induced by variation of ambient temperature is proposed. This correction is more specific to measurements in which the DTS instrument is subjected to significant temperature variations - typically outdoor field measurements with non field-customized DTS units. Finally, a thorough quantification of temperature uncertainty is proposed. The combined uncertainty notably includes uncertainty components from the calibration method (bottom equality condition. calibration bath temperature, linear regression), the correction for temperature oscillation and the built-in temperature averaging.

The three elements listed above are tested for a field application in a 800 m deep borehole equipped with a coaxial BHE. One fiber cable is placed in the annulus of the heat exchanger while another cable is placed in the center pipe. A single calibration bath is used for the two fiber cables.

The proposed calibration method leads to more consistent results than raw data, segregated calibration for each cable or on-site calibration using the instrument interface. The consistency is better between the two fiber cables – they show the same temperature under undisturbed condition and heat recovery – but also in time – the calibration leads to similar temperature profiles when performed in a different setup with about one year in between. Nonetheless, the different values for the differential loss between the two calibration at one year interval should be investigated further, even though they are not being strictly inconsistent according to the found uncertainty bounds.

The proposed correction of temperature drifts due to the influence of ambient air temperature variations successfully filters out oscillations from the raw data. The oscillations are filtered out for the whole test period even though the correction is only based on times at which there is a priori no reasonable correlation between the ambient air temperature and the temperature at the bottom of the borehole – typically, undisturbed condition or heat recovery without circulation.

At a given depth, the temperature uncertainty is consistent in time with 95% confidence bounds of ± 0.45, 0.58, 0.74, 0.90, 1.1, 1.3, 1.5 and 1.7 K at 100, 200, 300, 400, 500, 600, 700 and 800 m depth, respectively5. The temperature uncertainty increase non-linearly with depth, with the clearly dominant component from depth larger than 200 m being the parameter estimation through linear regression performed as part of the calibration. The contribution to the combined uncertainty from time-averaging, correction of temperature drift and bottom temperature equality constraint are negligible. The contribution from the bath temperature uncertainty, the parameter C and the Stokes/anti-Stokes are important at shallow depths6.

These results suggest that calibration should be in focus if one wants to reduce the temperature uncertainty for deeper boreholes (> 200 m). Perhaps a way to decrease the calibration contribution to uncertainty is to use more baths, as suggested in [39]. In particular, it would be interesting to investigate the use of a calibration bath at the fiber end to reduce calibration uncertainty. This can be achieve by splicing two fiber channels from the same cable – thereby obtaining a duplexed or double-ended configuration – or by using a single cable to measure the temperature profiles in the annulus and center pipe. Furthermore, the temperature equality prior at the BHE bottom could be kept in such configurations, which could also contribute to reduce the uncertainty (compared to a case without this constraint).

For more shallow boreholes, however, the uncertainty contributions from C and the bath temperature should also be considered when trying to reduce the temperature uncertainty. For the bath temperature, a well-mixed bath with precision thermometer can reduce uncertainty. As for C, this could achieve in similar way through another calibration bath that is better than the internal oven, i.e. where the bath temperature is more uniform and the bath temperature uncertainty is lower.

It is relevant to note that the proposed quantification of uncertainty partly includes potential temperature model errors (i.e. deviations from eqs.2-3). This is because model errors will show in the residuals which distribution is partly captured in the linear regression uncertainty. A case for which model error is not fully included is for instance when the variance of the residuals is dependent on the distance (in general any dependency of the residuals on $\mathbf{X}$ will invalidate eq.14).

The calibration method and quantification of uncertainty are applied to a specific site with unusual characteristics compared to other BHE applications. In particular, the large borehole depth of 800 m sheds light on the different uncertainty components. Applying the methods to another site would nevertheless be positive to test the method robustness. A lab test with independent temperature measurements would be even more relevant. The influence of the choice of distributions in the Monte Carlo study for the bath and bottom conditions should be investigated in future studies. The impact of spatially and temporally varying bath temperature might also be a relevant aspect to consider in such future studies.

One of the main points of quantifying temperature uncertainty with Raman-based DTS is the verification or validation of borehole heat transfer model. Although it is hard to invalidate a model with field data due to many uncontrolled parameters (e.g. ground water flow, exact borehole geometry) [59], DTS measurements could be very informative as to the model strengths and drawbacks. Better models lead to more accurate design and estimation of running costs. In turn, this would help improve the market penetration of environmentally-friendly technologies for heating and cooling (GSHP, BTES, ATES). Another relevant aspect is the determination of the geothermal gradient and related geothermal heat flux as highlighted in [47].

Besides the results strictly related to the DTS calibration and uncertainty, there are some interesting thermal features that deserve further analysis but that can nevertheless be mentioned here. The first one is the heat flux inversion that happens along the BHE depth that can be seen in Figure 5.c. At around 200 m depth, the calibrated temperature profiles in the annulus and the center pipe indeed cross each other, meaning no heat is injected (netto) below that point. This intersection moves down the BHE as time elapses under heat injection and reaches around 360 m at the end of the heat injection period. The heat flux inversion has been noticed in previous modeling work [60,61] for coaxial BHe with annulus as the inlet (which is sub-optimal for heat injection). A related question that arise is how much the noticed heat flux inversion would influence the result from a Thermal Response Test (TRT). After all if no heat injection occurs over a whole borehole section as is the case here, one may question if the test can provide any information regarding the thermal parameters within that section.

A second feature that can be noticed is the progressive homogenization of temperature under circulation. Using the mid-depth temperature as initial temperature uniformly applied in the ground is a longstanding assumption in long-term borehole heat transfer modeling [62]. The more interesting aspect here is perhaps the time it takes for the temperature profile to homogenize. Here after several hours of circulation, the temperature profile is not yet stabilized. In turn, one may wonder if a long stabilization time would have an impact on heat transfer modeling, by for instance leading to a different distribution of heat flux along the borehole.

Another notable feature is the temperature disturbances that seem to occur in the topmost 200 m section of the annulus (ch1). This could be due to laminar flow regime – since that the diameter is larger in that section (165 mm) than the rest of the borehole (140 mm) – or groundwater inflow, although none of these explanations is fully satisfactory [56].

5. Conclusions

DTS offer interesting perspectives in BHE and GSHP applications; among other things for field tests and heat transfer model validation/verification.

A new calibration method for and its associated quantification of uncertainty have been proposed in this paper for single-ended Raman-based DTS. The calibration method is based on a single calibration bath and uses temperature equality at the bottom of the BHE as an extra constraint. In addition a method to remove fictitious temperature drifts due to ambient air variations is suggested. The different methods are implemented for a case study with a 800 m deep coaxial BHE, in which a DTRT is conducted.

The calibration method and temperature drift correction show robust features and give adequate results. The quantification of uncertainty leads to 95% confidence bounds of ± 0.58, 0.74, 1.1 and 1.7 K at 200, 300, 500 and 800 m depth, respectively. The temperature uncertainty increases non-linearly with depth and is dominated by the uncertainty in the estimated parameters during the calibration process, for depths larger than 200 m. As for more shallow boreholes, the uncertainty contribution from the parameter C, the bath temperature and the Stokes/anti-Stokes also become important. All of these elements – the calibration process especially – should be focused on if the temperature uncertainty is to be reduced. Such distributed temperature measurements with quantified uncertainty are useful in determining the strengths and drawbacks of heat transfer models and to evaluate local thermal properties.

Besides the results strictly related to the calibration and uncertainty quantification, the paper offer also insights into thermal features that appears during the DTRT, namely a heat flux inversion along the borehole depth and the relatively slow temperature homogenization under circulation.