3.2 Validation of Methods

All methods proposed in Section §3.1 except the DAM1 are implemented as independent FORTRAN 90 modules, running under a Solaris environment. The DAM1 is not implemented due to technical reasons and its high similarity with the DPM1. Testing of the above methods using artificial temperature data with known soil thermal properties is utilised. This section gives an overview of the tests that have been performed in order to ascertain the validity, and compare the performance of the methods. Section §3.2.1 gives a short account of the data used for each experiment. The remaining sections present the results of the experiments.

3.2.1 Overview of Idealised Experiments

In order to test the methods for inverse determination of the soil apparent thermal diffusivity, idealised temperature data are generated for a period of 20 days. Using these temperatures as input data the DAM2, DPM1, DPM2, DNM, IHAM and IHPM calculate 24, and the INM calculates 4 values of soil apparent thermal diffusivity per day. A statistical evaluation is applied to the results. The mean bias is calculated

$$\bar{r}_n = \frac{1}{N} \sum_{n=1}^{N} \left( D_{n,calc} - D_{h,app,n} \right)$$ (3.14)

where $r_n$ refers to the $n$th residual and $D_{n,calc}$ is the $n$th calculated apparent thermal diffusivity. $D_{h,app,n}$ is the apparent thermal diffusivity used to generate to artificial temperature data for the $n$th time-step. Also, a measure for the degree of noise is calculated.

$$\tilde{\sigma}=\sigma(r_n-\bar{r}_n) = \sqrt{\frac{1}{N} \su... ...=1}^{N} \left( D_{n,calc} - D_{h,app,n} - \bar{r}_l \right)^2}$$ (3.15)

Note that $\bar{r}_l$ in Eq. (3.15) is independent of $n$. If $D_h$ is constant, dividing $\bar{r}_n$ and $\tilde{\sigma}$ by the apparent thermal diffusivity determines the relative bias and relative noise, respectively.

3.2.1.1 Noise (Experiment 1)

Soil temperatures are calculated every hour at depths of $10$, $20$ and $30 cm$ for $20$ days using Eq. (3.4) with $D_h=0.7 \frac{mm^2}{s}$, $T_0=10^{\circ}C$, and $T_\infty=20^{\circ}C$. Measurement errors are simulated by adding a pseudo-random real number taken from a Gaussian distribution with a zero mean and standard deviation $\sigma_T=0$, $0.0003$, $0.001$, $0.003$, $0.01$, $0.03$, $0.1$, $0.3$, or $1 ^{\circ}C$. The upper two levels of these generated temperature values are used to calculate apparent thermal diffusivities using methods DPM1, DAM2, DPM2, IHAM and IHPM. Methods DNM and INM use the temperature values at all three levels. The apparent thermal diffusivities determined by the DNM are averaged using floating averages over a $24 h$ period.

3.2.1.2 Temperature Shift (Experiment 2)

Soil temperatures are generated as in Experiment 1 with $\sigma_T=0.001 ^{\circ}C$^3.1. An offset of $0.2 ^{\circ}C$ is added to all temperature values at a depth of $20 cm$. This temperature dataset is processed as in Experiment 1.

3.2.1.3 Non-Sinusoidal Boundary Condition (Experiment 3)

Soil temperatures are generated every hour at depths of $10$, $20$ and $30 cm$ for $20$ days using Eq. (2.7) as a finite sum of the form

$$T(z,t) = T_{\infty} + \sum_{n=1}^{N} b_{n} e^{- \frac{z}{d_{n}}} \sin( \omega n t - \frac{z}{d_{n}} + {\phi}_{n} )$$ (3.16)

with $N=5$, $D_h=0.7 \frac{mm^2}{s}$, $T_\infty=20^{\circ}C$, $b_{n=1 \dots 5}=10.0$, $3.11$, $0.20$, $0.77$, $0.34^{\circ}C$ and $\phi_{n=1 \dots 5}=3.61$, $0.14$, $2.89$, $2.53$, $-1.32 rad$. These temperature values represent a typical day with clear skies. No artificial noise was added. This temperature dataset is processed as in Experiment 1.

3.2.1.4 Variation of $\Delta t$ and $\Delta z$ (Experiment 4)

Soil temperatures are generated with $4$, $12$, $24$, $48$ or $144$ samples per day at three depths, which are separated vertically by $\Delta z=0.05$, $0.1$ or $0.2 m$. The middle depth is located at $z=0.2 m$. They are calculated for a period of $20$ days using Eq. (3.4) with $D_h=0.7 \frac{mm^2}{s}$, $T_0=10^{\circ}C$ and $T_\infty=20^{\circ}C$. Measurement errors are simulated by adding a pseudo-random real number taken from a Gaussian distribution with a zero mean and standard deviation $\sigma_T=0.001 ^{\circ}C$. This temperature dataset is processed as in Experiment 1.

3.2.1.5 Various Values of Thermal Diffusivity (Experiment 5)

Soil temperatures are generated every hour at depths of $10$, $20$ and $30 cm$ for $20$ days using Eq. (3.4) with $D_h=0.05$, $0.1$, $0.2$, $0.4$, $0.8$ and $1.2 \frac{mm^2}{s}$, $T_0=10^{\circ}C$, and $T_\infty=20^{\circ}C$. Measurement errors are simulated by adding a pseudo-random real number taken from a Gaussian distribution with a zero mean and standard deviation $\sigma_T=0.001 ^{\circ}C$. This temperature dataset is processed as in Experiment 1.

3.2.1.6 Linear Moisture Increase (Experiment 6)

Soil temperatures are generated every hour at depths of $10$, $20$ and $30 cm$ for $20$ days using a numerical solution of the heat conduction equation as described in Section §2.3. The boundary conditions of the integration are chosen as in Eq. (3.2) and (3.3) using $T_0=10 ^{\circ}C$ and $T_\infty=20 ^{\circ}C$. Volumetric soil moisture is chosen to increase linearly over a period of $20$ days from $0.05 \frac{m^3}{m^3}$ to $0.5 \frac{m^3}{m^3}$. The corresponding soil thermal properties $c_h$ and $k_h$ are determined using Eq. (2.17) and (2.18), respectively. All other soil properties ( $\eta=0.535 \frac{m^3}{m^3}$, $\theta_q=0.393 \frac{m^3}{m^3}$, $\theta_c=0.042 \frac{m^3}{m^3}$, $\theta_o=0.029 \frac{m^3}{m^3}$) are chosen according to a real dataset (Foulum, §B.1). Measurement errors are simulated by adding a pseudo-random real number taken from a Gaussian distribution with a zero mean and standard deviation $\sigma_T=0.001 ^{\circ}C$. This temperature dataset is processed as in Experiment 1.

3.2.1.7 Idealised Rain Event (Experiment 7)

Experiment 7 only differs from Experiment 6 in the choice volumetric soil moisture. An idealised rain event is simulated using

$$\theta_w(t) = \left\{ \begin{array}{l@{\quad \quad}l} \theta... ...ac{t-t_{rain}}{\lambda} ) & t \ge t_{rain} \end{array} \right.$$ (3.17)

where $\theta_0=0.1 \frac{m^3}{m^3}$, $t_{rain}=5 d$, $\theta_{rain}=0.35 \frac{m^3}{m^3}$ and $\lambda=3 d$. The depth-dependent time-shift and decrease in amplitude of the water front, as well as changes in temperature due to infiltration are not simulated. Figure 3.1 shows the calculated soil moisture and apparent thermal diffusivities as a function of time. This temperature dataset is processed as in Experiment 1.

Figure 3.1: Artificial volumetric soil moisture and apparent thermal diffusivity for Experiment 7 as a function of time.

3.2.2 Sensitivity to Quality of Temperature Measurements

3.2.2.1 Noise (Experiment 1)

Table 3.1: Overview of the results from Experiment 1 for $\sigma_T=0 ^{\circ}C$. $\bar{r}_i$ and $\tilde{\sigma}$ are the mean bias and noise measure as described in Section §3.2.1. The values should be seen in the light of the reference diffusivity, namely $D_h=0.7 \frac{mm^2}{s}$ for Experiment 1. A value $\varepsilon$ indicates an error smaller than machine precision, approximately $10^{-15}$.

$\bar{r}_i$

$\tilde{\sigma}$

Method

$1 mm^2/s$

rel. Bias

$1 mm^2/s$

rel. Noise

DAM2

$\varepsilon$

$-$

$\varepsilon$

$-$

DPM1

$6.96$

$10^{-3}$

$1.0\%$

$6.09$

$10^{-2}$

$8.7\%$

DPM2

$\varepsilon$

$-$

$\varepsilon$

$-$

DNM

$-1.48$

$10^{-2}$

$-2.1\%$

$1.82$

$10^{-16}$

$2.6$

$10^{-13}$

IHAM

$\varepsilon$

$-$

$\varepsilon$

$-$

IHPM

$\varepsilon$

$-$

$\varepsilon$

$-$

INM

$-1.93$

$10^{-3}$

$-0.28\%$

$8.19$

$10^{-4}$

$0.12\%$

Table 3.1 summarises the results of Experiment 1 in the case $\sigma_T=0 ^{\circ}C$. The second order direct methods (DAM2, DPM2) as well as the inverse harmonic methods (IHAM, IHPM) are exact to within machine precision. With a relative bias from the reference value of approx. $- 2 \%$, the DNM already shows considerable deviation from the reference value for this highly idealised test. This is due to the fact that Eq. (2.4) cannot be solved for $D_h$ if $\partial_z^2 T$ is equal to zero. The relative bias of the DPM1 of approx. $1 \%$ is due to the limited accuracy in calculating the time of occurence of minima and maxima of the temperature curve. This is done using a smooth spline interpolation of the 24 temperature data per day. The INM underestimates $D_h$ by approx. $0.3 \%$. Smaller biases can be accomplished by increasing the accuracy of the heat conduction equation integration, resulting in time-consuming calculations.

Figure 3.2: Relative mean bias of apparent thermal diffusivities as a function of standard deviation of temperature data $\sigma_T$. See Experiment 1 for a complete description.

Figure 3.2 illustrates the relative mean bias as a function of the standard deviation of the artificial noise added to the temperature data. A complete illustration of the results for $\sigma_T=0.001 ^{\circ}C$ can be found in the Appendix (§C.1). Both the DPM1 and DNM show a considerably larger bias at small $\sigma_T$ than the other methods. At $\sigma_T \approx 0.03 ^{\circ}C$, the biases of the DPM1 and DNM increase rapidly. The DPM1 is very sensitive to an exact determination of the daily extremal temperatures. At high noise levels, several local extrema may occur and ambiguities in the determination arise. The failure of the DNM can be accounted for by numerical instabilities at certain data, which give a larger overall error. At $\sigma_T \approx 0.1 ^{\circ}C$, the biases of the higher order and indirect methods DAM2, DPM2, IHAM, IHPM and INM also exhibit strong deviations from the reference diffusivity.

Figure 3.3: Relative noise measure of apparent thermal diffusivities as a function of standard deviation of temperature data $\sigma_T$. See Experiment 1 for a complete description.

Figure 3.3 shows the relative noise measure $\tilde{\sigma}$ of the calculated apparent thermal diffusivities as a function of the standard deviation of Gaussian noise. At standard deviations smaller than $\sigma_T=0.1 ^{\circ}C$, all methods except DPM1 and INM show an approximately straight line in the log/log plot, thus implying a linear dependence of the noise measure on $\sigma_T$. The DPM1 and INM have asymptotic behaviours for $\sigma_T \to 0 ^{\circ}C$, due to the limitations in accuracy mentioned above. At errors greater than $\sigma_T=0.1 ^{\circ}C$, $\tilde{\sigma}$ exhibits a faster than linear increase for the DPM1, DAM1, DPM2, DNM. These methods no longer work reliably due to problems in the determination of daily temperature extrema and the numerical instabilities mentioned above.

3.2.2.2 Temperature Shift (Experiment 2)

Table 3.2: Overview of the results from Experiment 2. For a description of $\bar{r}_i$ and $\tilde{\sigma}$ see Tab. 3.1. The third and fifth columns compare the obtained $\bar{r}_i$ and $\tilde{\sigma}$ to the results of Experiment 1.

		$\bar{r}_i$							$\tilde{\sigma}$
Method		$1 mm^2/s$			$\bar{r}_i/\bar{r}_{i,noshift}$				$1 mm^2/s$				$\tilde{\sigma}/\tilde{\sigma}_{noshift}$
DAM2		$8.37$		$10^{-5}$		$1.0$				$6.19$		$10^{-4}$		$1.0$
DPM1		$6.59$		$10^{-3}$		$1.0$				$6.15$		$10^{-2}$		$1.0$
DPM2		$-8.49$		$10^{-5}$		$1.0$				$6.33$		$10^{-4}$		$1.0$
DNM		$-2.51$				$1.7$		$10^{+2}$		$9.93$		$10^{-1}$		$1.2$		$10^{+3}$
IHAM		$-8.70$		$10^{-5}$		$1.0$				$2.24$		$10^{-4}$		$1.0$
IHPM		$8.27$		$10^{-5}$		$1.0$				$2.68$		$10^{-4}$		$1.0$
INM		$-3.80$		$10^{-3}$		$1.9$				$2.47$		$10^{-2}$		$27$

The results of Experiment 2 are summarised in Tab. 3.2. Only the DNM and INM are affected by a systematic error in the temperature data. All other methods rely on the amplitude or phase of the daily thermal wave, which is not affected by an offset. The bias of the INM is within the expected range, whereas the large bias of the DNM is due to failure of the method at certain times. Using these two methods, care should be taken to avoid systematic errors of temperature measurements.

Experiment 1 and 2 both suggest that the DPM1 and DNM are not suitable for determination of apparent thermal diffusivities of the soil. Even with the highly idealised temperature data of the above experiments, they exhibit strong sensitivity to measurement errors. Due to these limitations, they will not be considered in the following experiments, with the exception of Experiment 3.

3.2.3 Sensitivity to Boundary Condition

3.2.3.1 Non-Sinusoidal Boundary Condition (Experiment 3)

Table 3.3: Overview of the results form Experiment 3. All tabulated quantities are explained in the caption to Tab. 3.1.)

$\bar{r}_i$

$\tilde{\sigma}$

Method

$1 mm^2/s$

rel. Bias

$1 mm^2/s$

rel. Noise

DAM2

$2.16$

$10^{-5}$

$3.08$

$10^{-5}$

$1.79$

$10^{-2}$

$2.6\%$

DPM1

$7.38$

$10^{-1}$

$105\%$

$1.24$

$10^{-1}$

$18\%$

DPM2

$1.42$

$10^{-3}$

$0.20\%$

$3.36$

$10^{-2}$

$0.48\%$

DNM

$-2.84$

$10^{-1}$

$-41\%$

$1.59$

$10^{-15}$

$2.27$

$10^{-15}$

IHAM

$\varepsilon$

$-$

$\varepsilon$

$-$

IHPM

$\varepsilon$

$-$

$\varepsilon$

$-$

INM

$-8.00$

$10^{-4}$

$-0.11\%$

$1.21$

$10^{-3}$

$0.17\%$

In Experiment 3, a non-sinusoidal boundary condition was chosen. Table 3.3 summarises the results for all methods. As compared to Experiment 1 ( $\sigma_T=0 ^{\circ}C$), only the inverse harmonic methods (IHAM, IHPM) are still exact to within machine precision. The performance of the DPM1 and DNM are not acceptable for non-sinusoidal boundary conditions, and therefore should not be used for real soil-data. The second order direct methods (DAM2, DPM2) seem to handle boundary conditions having more than two harmonics well, since higher harmonics are damped strongly. The noise measure of the INM method is approximately equal to Experiment 1, and is dominated by the precision of the heat conduction integration.

In natural soils, the diurnal temperature wave is almost never purely sinusoidal. Experiment 3 indicates that methods which assume sinusoidal boundary conditions, such as the DPM1, fail to give a satisfactory approximation of apparent thermal diffusivity with non-sinusoidal boundary conditions.

3.2.4 Sensitivity to Discretisation Intervals

3.2.4.1 Variation of $\Delta t$ and $\Delta z$ (Experiment 4)

Figure 3.4: Relative mean bias of apparent thermal conductivity $\bar{r}_i$ as a function of the number of temperature data per day. For a description see Experiment 4.

Experiment 4 consists of two tests. First, the time discretisation step, i.e. the number of temperature data available per day, is varied. In the second test, the vertical distance between the temperature data is varied. Figure 3.4 and 3.5 show the results of the first test. Both the bias and noise measure of the DAM2 and DPM2 are not sensitive to a variation of the number of available temperature measurements, since they always use four data. For a relative bias of less than $5\%$, the INM requires more than four samples per day. If $24$ or more measurements per day are available, all tested methods determine the apparent thermal diffusivity with a bias of less than $2\%$.

Figure 3.5: Relative noise measure of apparent thermal conductivity $\tilde{\sigma}$ as a function of the number of temperature data per day. See Experiment 4 for a description.

For the IHAM and IHPM, the relative noise measure (Fig. 3.5) decreases steadily with an increasing number of samples per day. The same is valid for the INM up to $24$ samples per day. At higher sample rates, the noise measure of the INM is restricted by the accuracy of the integration of the heat conduction equation. The DAM2 and DPM2 determine the apparent thermal diffusivity with an approximately constant relative noise of approximately $0.1\%$. If highly time-resolved temperature measurements are available, these methods do not give an increase in accuracy.

Figure 3.6: Relative mean bias of apparent thermal conductivity $\bar{r}_i$ as a function of the discretisation step $\Delta z$, as described in Experiment 4.

The results of the second test of Experiment 4 are summarised in Fig. 3.6 and 3.7. Biases of the DAM2, DPM2, IHAM and IHPM are small compared to the bias of the INM. For the INM, the relative bias shows a strong sensitivity to the space discretisation interval. The relative errors of the DAM2, DPM2 IHAM and IHPM decrease with increasing distance between temperature measurements. The apparent thermal diffusivity of the soil affects the amount of damping and phase shifting with depth of diurnal temperature waves. Hence, if the temperature data originate from levels far apart, the damping and shifting is more pronounced and signal-to-noise is larger.

Figure 3.7: Relative noise measure of apparent thermal conductivity $\tilde{\sigma}$ as a function of the discretisation step $\Delta z$. See Experiment 4 for a description..

The results of Experiment 4 seem to indicate that, with the exception of the INM, large space discretisation intervals are more suitable for a determination of soil apparent thermal diffusivity. Nevertheless, care should be taken when using temperature measurements from a soil. The results are somewhat misleading because natural variation of soil thermal properties and water content with depth are not considered in this experiment. A distance longer than $0.1 m$ can already lead to considerable error in the determination of the diffusivity. Knowledge of the soil composition profile at the measurement site is crucial for the ideal choice of $\Delta z$.

3.2.5 Variation of Thermal Diffusivity

3.2.5.1 Different Values of Thermal Diffusivity (Experiment 5)

Figure 3.8: Mean bias of apparent thermal conductivity $\bar{r}_i$ as a function of apparent thermal conductivity $D_h$. See Experiment 5 for a description.

Figure 3.8 and 3.9 illustrate the results of Experiment 5. Bias and noise measure are calculated as a function of the apparent thermal diffusivity. As a consequence of the different dependence of the damping and the phase-shifting on the apparent thermal diffusivity 1stsol, a different behaviour of the amplitude-dependent (DAM2 and IHAM) and the phase-dependent (DPM2 and IHPM) methods is expected. This can be observed in Fig. 3.8. While the biases of the DPM2 and IHAM do not vary very strongly with $D_h$, the DAM2, IHPM and INM show an increase in bias with increasing $D_h$ at values larger than $0.1 \frac{mm^2}{s}$.

Figure 3.9: Noise measure $\tilde{\sigma}$ of apparent thermal conductivity as a function of apparent thermal conductivity $D_h$. See Experiment 5 for a description.

The noise measure (Fig. 3.9) is generally larger for the second order methods (DAM2, DPM2). Due to stronger damping of high frequency noise at low apparent thermal diffusivities, all methods show a steady decrease in noise with decreasing diffusivity for $D_h>0.2 \frac{mm^2}{s}$. At very low diffusivities, the noise measures increase abruptly. Caution in the application of the above methods should be exercised. In particular the accuracy of the second order methods (DAM2, DPM2) can be affected considerably by measurements errors.

3.2.5.2 Linear Moisture Increase (Experiment 6)

In Experiment 6, the temperature data are generated by integrating the heat conduction equation subject to boundary conditions. Since this integration introduces additional sources of errors, results cannot be compared to the previous experiments. Results are compared to a test run using a constant soil moisture of $\theta_w=0.3 \frac{m^3}{m^3}$. The complete results of Experiment 6 are given in the Appendix (Fig. C.2). For the DAM2, DPM2, IHAM and IHPM a time-varying apparent thermal diffusivity introduces oscillations to the calculated diffusivities. Since these oscillations have a period of one day, averaging over $24 h$ eliminates the oscillations. The DPM2 and IHAM overestimate (underestimate) the apparent thermal diffusivity if $\frac{d D_h}{d t}$ is negative (positive), respectively. The DAM2, IHAM and INM always underestimate the thermal diffusivity. For the DAM2 and IHAM, this is also observed in the reference experiment, and is probably due to additional errors introduced by the direct integration of the heat conduction equation.

Table 3.4: Overview of the results from Experiment 6. For a description of $\bar{r}_i$ and $\tilde{\sigma}$ see Tab. 3.1. The third and fifth columns compare the obtained $\bar{r}_i$ and $\tilde{\sigma}$ to the results of a test run using a constant soil moisture of $\theta_w=0.3 \frac{m^3}{m^3}$

		$\bar{r}_i$							$\tilde{\sigma}$
Method		$1 mm^2/s$			$\bar{r}_i/\bar{r}_{i,ref}$				$1 mm^2/s$				$\tilde{\sigma}/\tilde{\sigma}_{ref}$
DAM2		$-3.04$		$10^{-3}$		$0.97$				$2.32$		$10^{-3}$		$3.8$
DPM2		$1.96$		$10^{-4}$		$-9.6$				$3.50$		$10^{-3}$		$6.2$
IHAM		$6.70$		$10^{-5}$		$2.9$				$2.05$		$10^{-3}$		$10$
IHPM		$-3.19$		$10^{-3}$		$1.0$				$1.29$		$10^{-3}$		$5.0$
INM		$-4.85$		$10^{-3}$		$1.0$				$7.59$		$10^{-4}$		$1.1$

Table 3.4 summarises the results of Experiment 6. The biases of the DPM2 and IHAM show a strong change when compared to the reference run. Nevertheless, they still have smaller biases than the other methods. The increase in noise measure for the DAM2, DPM2, IHAM and IHPM can be explained by the oscillations mentioned above.

The natural variation of the apparent thermal diffusivity due to a change in moisture rarely exceeds roughly $10\%$ of the absolute $D_h$ value. In order to give reasonable estimates of soil moisture, the methods for determining soil thermal diffusivities must be accurate to within at least one or two percent. Even when using idealised temperature data, biases and noise measures of some methods are considerable. Averaging over a period of $24 h$ can help reduce the error of the calculated diffusivities, but biases are difficult to eliminate.

3.2.5.3 Idealised Rain Event (Experiment 7)

Experiment 7 evaluates the response of the methods to a fast change in the volumetric moisture content of the soil. An illustration of the complete results of Experiment 7 is given in the Appendix (Fig. C.3). Since all methods use temperature data of one day to recover the apparent thermal diffusivity, the determined diffusivities resemble a $24 h$-average of the reference diffusivity. As in Experiment 6, strong oscillations occur when the diffusivity varies with time for all methods except the INM. Even with the abrupt change of diffusivity at $t=5 d$, the INM exhibits a very stable behaviour.

Table 3.5: Overview of the results from Experiment 7. All tabulated quantities are explained in the caption of Tab. 3.1.

		$\bar{r}_i$							$\tilde{\sigma}$
Method		$1 mm^2/s$			$\bar{r}_i/\bar{r}_{i,ref}$				$1 mm^2/s$				$\tilde{\sigma}/\tilde{\sigma}_{ref}$
DAM2		$-3.34$		$10^{-3}$		$3.7$				$2.32$		$10^{-3}$		$4.2$
DPM2		$-4.04$		$10^{-5}$		$2.0$				$3.50$		$10^{-3}$		$8.5$
IHAM		$-3.55$		$10^{-5}$		$1.6$				$2.05$		$10^{-3}$		$15$
IHPM		$-3.34$		$10^{-3}$		$1.1$				$1.29$		$10^{-3}$		$12$
INM		$-4.84$		$10^{-3}$		$1.0$				$7.59$		$10^{-4}$		$2.0$

Bias and noise measure are given in Tab. 3.5. The values reflect the description in the last paragraph. The DAM2, IHPM and INM underestimate the diffusivity by a considerable amount. As mentioned above, for the DAM2 and IHPM, this is artificially introduced by errors originating from the direct integration of the heat conduction equation.