The last preliminary task is to find the time range of the target signals. This time range is only necessary for the SotA methods and the SDMs, because the training-based FRM implicitly finds its useful time ranges through the feature selection. For simplicity and since the evaluation is based on the time-delay differences, whose ground truth is given by (6.1) only for the diametral path, only the direct propagation will be used. Because the algorithm to detect the time range needs to be very robust, several conditions are imposed on the algorithm to ensure that only meaningful time ranges are returned. However, a complete description of the algorithm would distract from the focus of the work, which is why only the basic approach is discussed here. Further details can be found in [157]. The basic principles of the algorithm can be summarized by the following steps:

Finally, the local maximum, which is closest to the theoretical time delay calculated by the path length and the SoS, is taken to be the wave packet of the direct propagation path. Based on this rough determination, the upper limit of the time range is the next consecutive local minimum of the envelope, as this is considered the time, when the next wave packet starts. Either the local minimum before the determined local maximum, the time at which the envelope falls below a certain threshold or the minimum limit, which depends on the upper limit and the determined local maximum, is selected as the lower limit of the time range. The decision which lower limit to choose is made in such a way that the resulting time range is as small as possible.

# 6.4 State-of-the-art methods

In this section the baseline for the proposed TDE methods is determined. For each data set, the time-delay differences of the direct propagation path are estimated by using the time-based zero-crossing method (ZCM), and four different correlation-like methods, namely the cross-correlation (CC), the generalized cross-correlation (GCC) with the phase transform processor, the AMDF, and the ASDF. Since the time-delay difference has to be estimated with subsample precision and depends on the observed time range, the SotA methods are combined with the time range detection presented in the previous section and with an interpolation. For the interpolation of the zero crossing, the method proposed by Kupnik et al. [63] is applied. In contrast, the CC, the GCC, and the ASDF use a quadratic interpolation around the maximum of the correlation function. Among the correlation-like methods, only the AMDF method is an exception, since here the objective is to find a minimum of the AMDF, whose shape looks like the absolute value of a sine wave around its minimum. Therefore, a double linear interpolation is applied for the AMDF-based subsample TDE.

The results of the ZCM, the CC and the AMDF for the four transducer pairs of data set D1c are shown in Fig. 6.5. As already expected, the estimation error shows an oscillation with a period in the range 1− 1.5 µs, which shows a good agreement with the theory as a period duration of

Figure 6.5 Error curves of some selected SotA methods. The respective MAPE is given in brackets. Although the estimation error of all measurement signals contained in the data set D1c is shown here, the evaluation metric MAPE was only calculated on the set covering one period cycle according to (6.8).

1/0 ≈ 1.4 µs was predicted theoretically. The different period durations for the different transducer pairs can be explained by the slightly varying material characteristics of each transducer. Only the local frequency of the target signals and the interfering signals in the evaluated time range are relevant. Therefore, the local frequencies may deviate from the excitation frequency. Furthermore, this figure shows that even the level of the interfering signals may differ from transducer pair to transducer pair, since, e.g., the amplitude of the oscillating estimation error [] for TP4 is significantly smaller than for TP3. Apart from that, all SotA

Table 6.2 MAPE results of the different SotA TDE methods for all data sets.

methods investigated here perform essentially the same with only slight differences. While the ZCM is best in terms of MAPE for two of the transducer pairs, the AMDF and the CC are better for the other two transducer pairs. Normally, the zero-crossing-based TDE would be more noisy, but using the approach of Kupnik et al. [63], the information of a whole period can be taken into account to get a zero crossing that is more robust against noise. For this reason, no obvious difference in the stochastic estimation error can be observed.

The MAPE results of the remaining SotA methods on all data sets are listed in Table 6.2. While the MAPEs of the respective best methods are highlighted in green, the average values over all transducer pairs are printed in bold. The best values of the respective data sets represent the baseline when, in the further course, the results of the SDMs or FRMs are given for individual transducer pairs or are averaged over an entire data set. It is noticeable that the damping mat has significantly reduced the interfering signals, because the MAPE values of the data set D2c are significantly lower compared to the data set without damping mat D1c.

Table 6.3 STDSPE results of the different SotA TDE methods for all data sets.

Furthermore, the ethanol data set D3,c seems to have the highest level of interfering signals, which can be explained by the worse acoustic coupling between the pipe wall and the liquid. Once again large differences can be observed between the different transducer pairs of the same data set. In summary, all SotA methods are in the same order of magnitude, with the GCC method showing slight advantages on average for all data set.

Complementary to the MAPE, which mainly evaluates the systematic error—except in the case when the systematic error is significantly smaller than the stochastic error—the STDSPEs for all data sets as a quality measure for the stochastic errors are given in Table 6.3. Here, it can be seen that, as already suspected from the error curves in Fig. 6.5, the stochastic errors of the methods are very similar, with the AMDF being slightly better than the other methods. Interestingly, the differencebased correlation of the AMDF can handle noise better, but against the interfering signals, it performs rather averagely.

# 6.5 Signal-dynamics method

The SDMs can be separated into the PCA-based, the B-spline(BS)-based and the joint-B-spline(JBS)-based point cloud processing, each of which can be further separated into the direct TDE estimation approach, the local interfering signals compensation and the global interfering signals compensation. The difference between the latter two is that in the local case, the interfering signals are compensated for each measurement package individually, while in the global case, all estimated interfering signals are combined, freed from outliers, and then averaged to get an estimate that is used for the interfering signals compensation of all measurement packages equally. Firstly, the direct TDE methods on a single data set are investigated, examining both the expected properties and the dependency on the process conditions. In the next step, the concept of compensating the interfering signals locally and globally is proven. Furthermore, visual examples of the estimated interfering signals, the outlier detection, and the globally estimated interfering signals are shown. This section concludes by examining how to specify the hyperparameters. To this end, the approaches are applied to the water data sets using all hyperparameter combinations. Then, using the optimized hyperparameter set, the performances of the SDMs are averaged over all data sets and TPs and compared with the GCC method—the best SotA method in terms of MAPE.

As in the rest of the thesis, for the SDMs to be applicable, the measurement signals of the data sets were divided into packages with a signal dynamic of 50 ns, i.e., within the package there are measurement signals whose absolute time delays cover an interval of the width 50 ns. An exception to this is the investigation of the influence of the signal dynamics in Section 6.5.3. Furthermore, the packages were selected from the entire data set with a variable overlap such that they uniformly cover the entire data set.

#### 6.5.1 Direct time delay estimation

In the first evaluation, the direct TDE methods are applied to the water data set with constant flow D1,c|TP1 to check whether the approach can also be applied to real data. After the signals were separated into

Figure 6.6 Results of the direct TDE using the SDM for the data set D1c|TP1. The signal dynamics of the packages are set to Δa = 50 µs. The median absolute time delay a, of each signal package was distributed uniformly on the whole process condition range.

packages and the relevant time ranges were extracted package-wise by the method described at the end of Sec. 6.3, the time-delay differences are estimated using the PCA-based, the BS-based, and the JBS-based point cloud processing, respectively. While for all methods the same signal dynamics (50 ns) were specified, the overlap for the PCA-based TDE was increased to get a densely sampled error curve comparable to the BS- and JBS-based SDMs. This is necessary since the PCA-based direct TDE can only estimate one average time-delay difference per package. The last hyperparameter to specify is the point cloud generation for the BS- and the JBS-based approaches. According to Equation (4.47), there are two possibilities: either the point clouds of the delay and reference signals are generated and processed separately, or the delay and reference signals are concatenated to get a combined point cloud. In this evaluation, the point clouds were concatenated.

The results are depicted in Fig. 6.6. Although all three methods basically estimate the time-delay difference well, there are slight differences, which can be explained by the different properties. The PCA-based SDM is less noisy. This is to be expected because the time-delay differences are averaged over the whole signal package, which can be interpreted as an individual estimation with subsequent moving average filtering. Furthermore, the joint estimation of the BSs for two consecutive time steps obviously also reduces the stochastic component of the TDE. However,

Figure 6.7 Results of the direct TDE using the SDM for the data set D1v|TP1 with varying time delay. The signal dynamics of the packages are set to Δa = 50 µs. The performance of the PCA-based SDM deteriorates significantly because the requirement—a constant time-delay difference—is not met at all.

in terms of systematic error, the BS-based SDM shows better results than the PCA-based and the JBS-based SDMs. The larger systematic error of the PCA-based SDM can be explained by the fact that the constant time delay assumption does not perfectly hold, since the time-delay difference varies even in constant flow scenarios due to the varying SoS (see Fig. 6.3). Contrary to this, the JBS-based approach relies more on the assumption that the interfering signals are stationary, i.e., they do not vary with varying temperature or varying VoF, compared to the BS-based SDM. This requirement is also not quite met, which is why probably the JBS-based SDM shows a slightly higher systematic error.

If the methods are evaluated for a data set with varying flow D1,v|TP1, as shown by the results in Fig. 6.7, the situation is quite different. While the spline-based methods still work well, the PCA-based SDM is no longer applicable because the requirement—a constant time-delay difference is not met at all. Contrary to the PCA-based SDM, a visual comparison of the spline-based methods with each other is hardly possible in this presentation form. Only for high absolute time delays, corresponding to low temperatures in the water data sets, where air bubbles are more often present in the experimental setup, the JBS-based approach shows more outliers, but further evaluations are needed for a general statement.

Figure 6.8 Estimated interfering signals for the data set D1c|TP1. The different interfering signals result from different signal packages, which were created using a signal dynamic of 50 ns with an adaptive overlap as described in the introduction to Section 6.5.

#### 6.5.2 Interfering signals compensation

In the alternative approach, the interfering signals are first determined and removed, and then the time-delay differences are estimated using a state-of-the-art (SotA) method, instead of estimating the time-delay differences directly. As already mentioned in Section 4.2.3, the ZCM according to Kupnik et al. [63] is used for the subsequent TDE after the interfering signals compensation. In Figure 6.8 all estimated interfering signals are shown that result when the interfering signals are estimated separately for each of the signal packages. Equally to the evaluation of the direct TDE, the data set is D1c|TP1, the signal dynamics are set to 50 ns per package, the point clouds are concatenated for the splinebased approaches, and the PCA-based SDM uses the constant amplitude assumption. Although most signals are estimated to be quite similar for all methods, some differences occur that cannot be explained by the nonstationarity of the interfering signals. The JBS-based approach, which is expected to be the most robust method, shows the least variation of the estimated interfering signals. Therefore, it is likely that the reasons for the variation are the noise influence and outliers in the point clouds that deteriorate the individual BS-based shape fit of the point cloud. Furthermore, an incorrect assumption of the signal model, e.g., that the amplitude of the target signals does not depend on the temperature, could be the cause.

Figure 6.9 TDE results using the local interfering signals compensation for the undamped water data sets with constant and varying VoF. Overlapping TDEs between the signal packages are resolved by averaging.

If each signal package is compensated by the respective estimated interfering signals, application of the zero-crossing-based TDE yields the time-delay differences for all signals in the package—this is called the local interfering signals compensation. Any overlapping estimates resulting from the overlap of the signal packages can be resolved by averaging. The results for the data sets D1c|TP1, D1v|TP1 can be observed in Fig. 6.9(a) and Fig. 6.9(b), respectively. In Fig. 6.9(a) it can be seen that the PCA-based SDM continues to perform well for the constant flow data set. However, the BS-based point cloud processing does not work in combination with the local interfering signals compensation (red curve). The inconsistency at 69.7 µsis caused by a gap in the measurement signals, which results from the automatic outlier removal of signals degraded by bubbles in the experiment. Figure 6.9(b) shows that the PCA is still not comparable to the spline-based methods for varying flow data sets. This is due to the fact that the abrupt changes of the flow lead to noncompact point clouds, which cannot be correctly processed by the PCA. In contrast, the spline-based methods show that they are suitable for data sets with highly varying flow velocity because the curved form of the BSs is flexible enough to fit even point clouds with large gaps.

Figure 6.10 Automatic outlier detection to remove bad estimations of interfering signals from Fig. 6.8. For better visualization, an offset has been added to the results from the PCA and the JBS-based method. Left: the detected outliers are shown in dots. On the right, the final estimated interfering signals can be seen, resulting from averaging after outlier removal.

In the global interfering signals compensation, for each signal package the same interfering signals estimate is used for compensation. This can reduce the effect of bad estimates of the interfering signals, which result from outliers in some of the point clouds. To this end, the different estimates are collected and outlier signals are removed by applying the outlier detection algorithm proposed in Section 4.4. The remaining estimates for the interfering signals are averaged to obtain the globally valid estimate—in the following, using this globally valid estimate will be referred to as global interfering signals compensation. A visual example of this procedure can be observed in Fig. 6.10. On the left side, all estimates for the interfering signals are plotted for each point cloud processing method—an offset has been added for better visualization. While the remaining estimates after the outlier detection are drawn solid, all signals classified as outliers are drawn dotted. It is easy to observe that due to the adaptive histogram the limit that defines which signals are identified as outliers is dependent on the variability of the estimates. In the case of lower variability, such as for the JBS-based SDM, even small deviations lead to a classification as outlier, while a higher overall variability, such as for the PCA- or BS-based SDM, does not classify signals as outliers even with larger deviations. The averaged results are then depicted on the

Figure 6.11 TDE results of the global interfering signals compensation for the undamped water data sets with constant and varying VoF, resulting if all measurement signals are compensated by subtracting the same averaged interfering signal.

right side. It shows that the resulting averages of the BS- and JBS-based SDM are quite close, leading to the expectation that the subsequent TDE based on this estimate will perform similarly.

Application of the global interfering signals compensation to the TDE for the water data sets D1c|TP1, D1v|TP1 yields the estimation errors depicted in Fig. 6.11. Through the usage of a global estimate for the interfering signals, the SDM now produces similar results for the constant and varying flow data sets. Only slightly increased stochastic components at measurements with low time-delay differences can be observed (see the small ripples in Fig. 6.11(b)). Moreover, the expected similar behavior of the BS- and the JBS-based SDMs is confirmed. An important point to notice is that even a small phase shift in the estimated noise signals (Fig. 6.10 on the right) can lead to large estimation errors. This is shown by the larger errors of the PCA-based SDM in Fig. 6.11, whereas the interfering signals estimate exhibited by the PCA-based SDM is only slightly phase-shifted compared to the BS- and JBS-based SDM.

In summary, each of the SDMs has its own advantages and disadvantages. The PCA-based point cloud processing, regardless of whether direct TDE, local or global compensation is used, can only be applied with sufficient estimation quality to constant flow data sets. For data

sets with varying flow velocity, the spline-based methods are generally better. In addition, the spline-based SDM using the global interfering signals compensation has been shown to have the lowest estimation error, regardless of whether the method was applied to a constant or a varying flow data set.

### 6.5.3 Hyperparameter optimization and overall results

The previous two sections only represent a proof-of-concept of the SDMs with a standard set of hyperparameters using the water data sets as an example. While the results could be used to make simple statements about the different approaches, they were neither evaluated quantitatively, nor tested on all data sets with all hyperparameter combinations, nor compared to the SotA methods. Therefore, in this section, the SDMs are firstly tested for all hyperparameter combinations, where a focus lies on the available signal dynamics and their interaction with the available SNR. Finally, all SDMs in their best hyperparameter configurations are applied to the water and ethanol data sets, evaluated in terms of MAPE, and then compared to the best SotA method—the GCC.

To begin with, the different hyperparameter combinations are tested. Figure 6.12 shows the performance of all SDMs in the form of a star plot where the categorical hyperparameters are varied for the water data sets D1c|TP1, D1v|TP1. A list of the investigated hyperparameter sets are found in Table 6.4. Note that for the PCA-based SDMs only the individual point cloud generation is possible since the geometric

Table 6.4 Combinations of hyperparameters available in the SDMs.

Figure 6.12 MAPE results of all three major point cloud processing methods in direct, local compensation and global compensation form for both data sets D1c|TP1 and D1v|TP1. The MAPE is coded as the distance from the origin for D1c|TP1 and D1v|TP1 in green and blue, respectively. Each angle in the star graph represents a hyperparameter configuration of the respective processing method. The baseline in its best configuration (best SotA method) is drawn dashed. For better visualization MAPE values are capped to 7.5 %.

interfering signals estimation requires the PCA of two distinct point clouds to triangulate the interfering signals. Since the computational effort to calculate the hyperparameter combinations full factorial is too high, the signal dynamics are fixed to 50 ns in this first hyperparameter investigation. The subsequent signal dynamics evaluation will show that this setting is justified. Furthermore, the BS- and JBS-based SDMs share the same hyperparameter sets, denoted by BS 1/2, because they only differ in the way the splines and the misalignments are estimated, but not in the way the point clouds are generated. The results from Fig. 6.12 once again shows, that the constant flow data set is a lot easier to manage and that the PCA cannot be applied in the varying flow case with sufficient

Figure 6.13 Interaction between used signal dynamics Δa and measurement noise for the data set D1c|TP1. Two different noise levels were examined: the original SNR (85.3 dB, solid lines) and the deteriorated SNR by adding artificial AWGN (62.9 dB, dashed lines).

accuracy. Nevertheless, the varying amplitude assumption ( PCA 3 ) shows slightly better results compared to the constant amplitude assumption, which is why it will be used in the following investigations for the PCAbased SDMs. For both spline-based approaches, the concatenated point clouds are preferable on average. However, even if the hyperparameter combination BS 1 is used, both spline-based methods are still better than the baseline, except for the direct estimation with individual point clouds.

Based on these hyperparameter configurations, the influence of the available signal dynamics is investigated. To this end, the evaluation of each method on the data set D1c|TP1 is repeated with different signal packaging strategies, where the package size was adapted such that the signal packages contain the specified signal dynamics. Starting from the minimum 20 ns, the signal dynamics were increased in steps of 5 ns, 10 ns, and 20 ns, depending on the required computational effort of the

respective method and the expected variation in the results—for higher signal dynamics the step size was consequently increased. Figure 6.13 depicts the results of this investigation. Since the graphic examines several aspects of the different SDMs, the results are interpreted step by step.

Firstly, the MAPE representing the systematic error is investigated. As can be easily seen, except for the BS-based SDM using the local compensation, the systematic errors are not significantly reduced further for signal dynamics greater than 50 ns. The JBS-based SDM would even allow lower signal dynamics with only minor degradation of the MAPE. To investigate the relationship to the available SNR, artificial AWGN is added to simulate a lower SNR. The MAPE curve for the lower SNR (drawn as dashed line) shows that the lower limit of the signal dynamics, below which the SDMs can no longer be used with sufficient quality, shifts towards higher signal dynamics. Furthermore, the lower robustness of direct TDE is revealed, which is concluded from the fact that the MAPE varies much more with different signal dynamics and requires larger signal dynamics to perform comparably. Summarizing the dependence of the systematic error on the available signal dynamics, it can be said that the influence is small as long as the signal dynamics are above the lower limit, which is SNR dependent.

Secondly, the STDSPE representing the stochastic error is investigated. Except for the direct estimation form of the SDMs, the stochastic error is again hardly dependent on the available signal dynamics. Similar to the systematic error, the lower limit of the signal dynamics, below which the SDMs can no longer be used, increases. However, additionally to the influence on the lower limit, a lower SNR leads to an increased offset of the stochastic error. The only exception to these statements is the direct estimation. Due to the averaging effect, the PCA-based direct estimation SDM can reduce the STDSPE with increasing signal dynamics, but as already mentioned this is bought by an increased response time of a measurement system based on this algorithm. The other spline-based SDMs do not show that they can reduce the stochastic error below the thresholds approached by the local, and global compensation methods. In summary, as long as the signal dynamics are higher than the SNRdependent limit, further increasing the signal dynamics provides no

Figure 6.14 MAPE performance of all SDMs averaged over the different TPs. The results achieved with the best SotA method is drawn as a dashed line. The error bars indicate the respective best and worst results received if the TPs are investigated individually.

advantage, except for the direct estimation methods, since they are less robust to noise.

Based on the hyperparameter influences presented above, the final overall evaluation of the SDMs is done using signal dynamics of 50 ns, the constant amplitude assumption for the PCA-based SDM in its interfering signals compensation variants, and the individual point cloud generation for the spline-based SDMs. Since the systematic error is more significant in the scope of this thesis due to its correlation with the level of the interfering signals, the results are compared in terms of MAPE in Fig. 6.14. Here, the averaged MAPE is depicted for the water and ethanol data sets without damping mat. Averaged MAPE means that the value is determined by averaging the MAPEs of the four transducer pairs per data set. Therefore, the basis for comparison—the performance of the GCC—is also averaged over all four respective transducer pairs. Note that the water data set with damping mat is not included in this evaluation since the temperature-dependent damping violates the assumption of stationary interfering signals. The results show that the PCA-based SDMs cannot be successfully applied to data sets with a strongly varying flow. In contrast, using the local or global compensation, the JBS-based SDM is significantly better than the best SotA for all data sets. Especially, the global compensation works well for both constant and varying flow data sets. Although the BS-based SDM is also often better than the GCC, some combinations, such as the direct estimation for D1c, or the local compensation for D1,v, perform worse. The last point worth mentioning here is the range of systematic errors between the transducer pair with the largest and smallest MAPE. Some SDMs can significantly reduce this range, e.g., the JBS-based global compensation reduces the initial MAPE range for the ethanol data set D3c from about 11 % to about 3 %.

In summary, except for the PCA-based and some variants of the BSbased SDM, all SDMs perform significantly better than the respective best SotA method in terms of averaged systematic error. Furthermore, the interfering signals compensation does not induce an increased stochastic error.

### 6.6 Feature-based regression method

Unlike the SDMs, the FRM requires a training data set, but it also allows to obtain an estimate for each signal individually without having to separate the signals into packages with some signal dynamics. Since the training set should contain varying time-delay differences and different process conditions, such as the temperature, the training data set for this approach is always chosen from the varying flow data sets. Accordingly, the corresponding constant flow data sets, whose time-delay differences are not included in the training data set, are used as test data sets to predict the time-delay differences of measurement signals that were not

seen by the model during the training steps. For a compact representation, a simple notation is introduced, to indicate which data sets are used as training and test data set, respectively. Exemplary, the notation D1v|TP1/D1c|TP1, denotes that the first TP of the varying flow data set D1v is used for training and the corresponding TP of the constant flow data D1c set is used to test the estimation performance.

However, due to the large number of options to select the features, the available regression models and their training strategies, the FRM requires the specification of much more hyperparameters than the SDM, where only the signal dynamics and the point cloud generation are important. A full factorial evaluation takes to much computational effort. For this reason, starting bottom-up from the regression model, the hyperparameters are investigated consecutively and then fixed to their respective best choices in the first two subsections.

Subsequently to the determination which regression model, training strategy, and feature selection method to use, the properties and performances of the FRM are evaluated in different scenarios. Firstly, the robustness against AWGN is tested, followed by an investigation if frames—in this case an overcomplete AWPT tree structure—can further improve the results. Finally, the overfitting and the generalizability are evaluated, since these properties are highly important in learning-based approaches.

All performances of the FRMs in this section are expressed in terms of the MAPE (6.7) for the systematic and the STDSPE (6.10) for the stochastic error in order to reduce the entire evaluation of a specific hyperparameter set to a single value.

#### 6.6.1 Selection of the best regression model

To begin with, the different regression models are investigated. However, since the evaluation of the FRM depends not only on the regression model but also on the feature selection, the regression models are evaluated according to how well they perform on average when using different feature selections. Thereby, not all feature selections can be considered in the average evaluation, because there are too many combinations of feature selections possible, i.e., the quality measure (F-test, PCC, …), the SS or the IFS, and the number of remaining features or thresholds can

Table 6.5 Average MAPEs using the three different regression models and their corresponding training strategies. The averaged MAPE values are obtained by using different numbers of selected features from s = 10 … 100. Here, the data sets D1v|TP1/D1c|TP1 are used for the training and the test.

be combined almost arbitrarily. Therefore, the type of feature selection is fixed to the ISS where the parameter of the remaining features s is varied.

According to sections 5.4 and 5.5, a complete hyperparameter set for the FRM contains the following parameters:

Additionally, in order to apply the method, the data sets for training and testing also need to be specified.

Table 6.5 shows the resulting average MAPEs of all available regression models and training strategies, if the FRM is applied multiple times with a differing number of selected features s and the individual MAPEs are averaged. Note that the parameters of the GPR cannot be trained using the least-squares or median-based approaches, so simple kriging is used here to fit the GPR. To simultaneously evaluate the dependence of the results on the number of features s , the standard deviation over the results of the different feature numbers is given in round brackets. Comparing the GPR with the linear regression approaches, it performs worst in the supervised case and only average in the unsupervised case. Therefore, and due to the fact that it requires the highest computational effort, this regression approach is not pursued further. Furthermore, it is noticeable that while the LS training yields the best results for the supervised FRM, the performance in the unsupervised FRM is significantly worse, suggesting that this type of training is less robust to errors in the labels. In contrast, the weighted training methods (WLS, WM) show similar behavior for both the true labels and the estimated labels. A comparison between the regression models indicates that the decision whether the model should include an offset or not is only relevant for the WM training. The application of this strategy leads on the one hand to the best results on average, if combined with the linear model with offset (5.35a), and on the other hand to nearly the worst results on average, if combined with the linear model without offset (5.35b). Observing the standard deviation shows that the dependence on the number of selected features is similar for all weighted training strategies, independently of the regression model. Only the LS method is significantly less dependent on s .

In summary, the use of Gaussian processes for regression is not appropriate for the regression of ultrasonic TDE in this algorithmic context. Since the WM method in combination with model (5.35a) performs best on average and the LS method has both the least dependence on the number of selected features s and the best results for the supervised FRM, these two strategies are further investigated. Here, the LS method is used without offset, because the model without offset physically makes more

Table 6.6 Evaluation of different methods to select the subset of features to be used for the supervised regression. The averaged MAPE values are obtained by using different numbers of selected features from s = 10 … 180. The regression model (5.35a) was chosen and trained using the WM strategy. As training set and test set, D1v|TP1 and D1c|TP1 were used, respectively.

sense when the features are designed to yield zero without time-delay difference.

#### 6.6.2 Feature selection evaluation

After setting the regression model to the linear model trained by the WM or LS method, the different feature selection methods can be quantitatively evaluated. Given the stable results for both supervised and unsupervised FRM, all available feature selection methods are tested by keeping the regression fixed to the linear model (5.35a) with the WM-trained parameters. Again, the different FRM setups are applied exemplarily to the water data sets D1v|TP1/D1c|TP1 and the results are averaged over different top-N values s to become invariant to this hyperparameter.

The results of this evaluation are listed in Tab. 6.6. For all methods, except for the ISS, where only the subspaces can be selected and not the feature individually, the MAPE is given once for the SS and once for the IFS. Among the tested methods, only the ISS, the NCA, and the Laplacian scores allow feature selection without knowledge of the labels. Therefore, these methods are preferable if the MAPEs are comparable.

Figure 6.15 Dependence of the best five feature selection methods on the number of selected features s , if the WM parametrized linear regression with offset is applied. The regression is trained with the ground truth for the labels.

Since the RReliefF, the wrapper method, and the Laplacian and F-test scores perform significantly worse on average, only the ISS, the PCC, the NCA, the LASSO, and the GPR are further investigated. An interesting fact is that, although the Gaussian processes were not suitable as a regression model, the learned length scales in the kernel function could be used well as a quality measure for the feature selection. The last point that can be seen from this study is that—for the feature selection in this specific learning-based TDE approach—the simple PCC is the best supervised method and the ISS proposed in this thesis is the best unsupervised method.

In order to determine the number of selected features that yields the best overall results, the performances of the five best feature selection methods with respect to MAPE and STDSPE are plotted against s in Fig. 6.15. While the IFS using the NCA or the LASSO becomes unstable at high numbers of features, the SS is stable for all methods and all s . However, even though the IFS is unstable for some methods, there are also combinations, such as the PCC or the LASSO for a low number of features, that result in the best MAPEs that can be achieved overall. Since the feature selection should yield a good robustness against noise, a low systematic error, and as few selected features as possible to save computational effort, the number of features will be fixed to s = 100 in the following studies. Figure 6.15 shows that using 100 features is a good compromise between the objective to have few features and high robustness against noise (see the subfigure at the bottom on the left side). The LASSO and the NCA selection methods are not considered further, because in some configurations their systematic error is too high. Furthermore, the GPR is also discarded due to the significantly larger computational effort with only comparable results.

#### 6.6.3 Robustness against additive white Gaussian noise

In the previous section, the many possible combinations of feature selection methods, regression models, training strategies and label choices for the training were evaluated and reduced to a few remaining possible combinations. The FRM with one of these remaining configurations can now be investigated regarding robustness against noise. Since the unsupervised FRM is most comparable to the SotA, which also assumes no known labels, the features are selected by the ISS (s = 100) and the linear regression model trained with estimated labels (and the WM strategy) is applied to the TDE. Although the present SNR cannot be further increased, the performance of the FRM can be investigated for decreased SNR by adding artificial AWGN to the training data set, the test data set, or both.

To this end, the water data sets D1v|TP1/D1c|TP1 are selected and deteriorated by adding noise. If the level of added AWGN is varied and the results of the FRM are plotted over the respective noise level, the diagram in Fig. 6.16 is obtained. Depending on which data set the AWGN is added to or if the data is smoothed by a prefilter, the different curves result. Note that the MAPE of the baseline (AMDF) is not drawn, since its

Figure 6.16 Impact of AWGN in the training and test data on the regression accuracy. Left: Influence on the systematic error evaluated by the MAPE. Right: Influence on the stochastic error evaluated by the STDSPE. The level of the added noise is specified by , where the subscripts (⋅)Tr and (⋅)Ts denote whether the noise is added to the training data set, the test data set, or both. For reference, the STDSPE of the best SotA method in terms of robustness against AWGN (the AMDF) is given.

systematic error is higher than the visible range. Some interesting effects can be observed in Fig. 6.16:

here, the systematic error of the AMDF method is almost independent of the added noise level, leading to the fact that the FRM is only better than the SotA if the SNR during training is sufficient.

#### 6.6.4 Application of frames in the feature generation

As the evaluation of the different feature selection methods and the dependence on the hyperparameter s has shown, the features, which are available for the regression, are an important influencing factor. Therefore, not only the selection process is relevant, but also the initial set of all features. Remembering Section 5.3, the features used in the regression are created by the absolute and phase differences of the coefficients that result from the AWPT of the measurement signals. However, the AWPT also requires the filter types and tree structure to be uniquely defined. Since the two preliminary studies [151, 159] have shown that the 6th level of the AWPT tree provides a good compromise between time and frequency resolution, all previous FRM evaluations are based on the coefficients of the 6th level in the tree. Nevertheless, the validation of this choice is verified again in this section. To this end, the set of available initial features is extended by using all coefficients of the 5th, 6th and 7th level in the AWPT to generate the features. This has the special effect that the subspaces covered by the coefficients are now overlapping in the time-frequency domain, which induces a redundancy in the representation—also called frame. Furthermore, the information is redundant, since, e.g., the coefficients from the 7th level contain all the information that is already contained in the coefficients from the 6th level.

The bar graph in Figure 6.17 visualizes the MAPEs—averaged over all TPs and the respective best and worst results—for the differently configured FRMs both when using only the 6th level coefficients and when using the redundant coefficients of the 5th, 6th and 7th level, which build a frame. In order to make the set of selected features comparable, the number of features is set to s = 100 for all selection methods. Except for the ISS in combination with the LS regression and estimated labels, using frames yields similar or worse results. There are several possible explanations for this behavior, which have to take into account the different characteristics of the feature selection methods, the choice

Figure 6.17 Average MAPEs over all TPs of the data sets D1v/D1c when using the 6th level AWPT coefficients (W = 6) to generate the features, and when using all coefficients of the 5th, 6th and 7th level (W ∈ {5, 6, 7}) to generate the features. The error bars indicate the respective best and worst result received if the TPs are investigated individually. For all feature selection methods, s = 100 applies.

of the labels, and the training strategy of the regression. Firstly, the PCC is considered, where the features are selected according to how well they are linearly correlated with the ground truth. Here, the results are very similar regardless of regression training or selection strategy, which in this context means, whether to use SS or IFS. This suggests that the selected features of the 6th level already contain the best features, confirming the validity of the initial choice of coefficients for the feature generation. In fact, a detailed examination of the resulting selection masks (not shown here due to the poor visual evaluability) reveals that 32 features between the initial feature set with W = 6 and with W ∈ {5, 6, 7} are equal.

Next, the ISS is considered, where after several iterative selection steps, the features with the largest variances are selected. Here, the application of frames for the feature generation leads to worse results when the WM regression is used, and to better or similar results when the LS regression is used (cf. the red and blue bars with the ISS labels in Fig. 6.17). Two interesting conclusions can be drawn from this behavior. Firstly, even though the LS regression is not very robust to errors in the labels, as

shown in Sec. 6.6.1, it can better balance different feature selections by suppressing features with low or poor influence in regression. This statement is based on the fact that the MAPEs of the LS-based regression with ISS and supervised training are significantly better than the MAPEs of the WM-based regression. Secondly, the ISS returns a worse performing selection of features when the initial feature set is larger due to the frames. Obviously, the objective to search for the maximum variance cannot perfectly represent the quality measure that as much information as possible should be contained in the selected subspaces.

The conclusion of this investigation is that in the problem at hand the use of frames does not improve the performance, since a sufficient number of good features can already be derived from the coefficients of the 6th level given the sampling rate 50 MHz.

### 6.6.5 Generalizability of the learning approach

Since it is the nature of learning-based methods that the quality of the training data set significantly affects the performance of the prediction, this section presents a quantitative assessment of the generalizability of the FRM. The effect that the systematic error on the training data set is much lower than that on the test data set is called overfitting. It indicates that the model may be too complex and, therefore, can perfectly interpolate the training data but hardly predict values not contained in the training data. The opposite of this, which means that the performance on the test data set is comparable to the performance on the training data set, is called generalizability.

For this reason, first, the performance on the test data set is compared with the performance on the training data set, and second, the resulting MAPEs are examined when the training data is restricted to include only a subset of the process conditions.

#### Performance on training data set vs. test data set

Figure 6.18 shows both the MAPE of the approximation of the trained FRM on the training set and the prediction error on the test set. In all selection and training scenarios (unsupervised/supervised), the performance on the training data set is worse than or similar to the performance

**WM regression with offset**

Figure 6.18 Average MAPEs over all TPs with the WM-trained linear regression model (5.35a). Blue bars: the MAPEs calculated on the training data set D1v. Red bars: the MAPEs calculated on the test data set D1c. From left to right, the hyperparameter combinations for feature selection and the used labels are plotted. The error bars indicate the respective best and worst results received if the TPs are investigated individually. For all feature selection methods, s = 100 applies.

on the test data set. While the MAPEs are quite similar for the PCC-based selection, the ISS yields a smaller systematic error on the test sets. From these results, two conclusions can be drawn. Firstly, the linear regression model is probably too simple to accurately approximate the time-delay differences on the training data set. This can be explained by the fact that the time-delay differences of the training data set are highly varying and have three clustering points due to the design of the experiment. Both properties make the training data set more complex than the test data set. Secondly, the PCC-based selections yield in general better feature sets, which can also be applied to the estimation of time delays that are closer to the boundaries of the process conditions that the FRM was trained with. However, this is to be expected since the PCC directly ranks the features according to their correlation with the labels and the ISS can only make use of quality measures that work without knowledge of the labels, such as the signal energy or the variance. Whether the ground truth is used for the training of the regression or the labels are estimated using the optimization approach seems to have no influence in this investigation.

#### Generalizability to different process conditions

In contrast to the investigation of the MAPE difference between the training and test data sets, the influence of constraining the training sets is investigated in Fig. 6.19. The aim is to evaluate the extent to which the FRM can be used for TDE if the process conditions of the test set are outside the process conditions used for training. To this end, the MAPEs are given for the subsets of the test data set, which only contain process conditions that are also included in the training data set, and for the entire test data set. In the following, the MAPEs on the subsets of the test set, which contain only the process conditions present in the training data set, are referred to as local MAPEs whereas the MAPEs on the entire test set are referred to as global MAPEs. Note that, if range{t }/range{g } = 1, the process conditions of the training and test data set are identical, resulting in the local and global MAPEs yielding the same values. Furthermore, the results show that the LS-trained regression is less generalizable to other process conditions. This can be inferred from the global MAPEs, which demonstrate degradation in performance relative to the respective best SotA method if the training data set is constrained to contain less than half of the available absolute time delay variations. In contrast, the local MAPEs are equally good for all regression methods. Note that the scale of the plots for the LS-based regressions goes up to 9 % due to the larger systematic errors. An interesting fact is that the unsupervised WM-trained regression shows little dependence on the available training set. This can be seen from the behavior of the average and standard deviation of the MAPEs, which show a similar magnitude regardless of the available training data set.

In summary, the supervised WM-trained regression generalizes best to different process conditions. Furthermore, even though the unsupervised WM-trained regression has a higher variance, it can be applied with only little degradation if the available training set contains less varying process conditions. However, note that WM-trained regression is better than the best SotA method—here the GCC—regardless of the available training set or whether the labels used for training are the ground truth or estimated by the approximation approach.

Figure 6.19 Performance of the FRM on the test data set D1c|TP1 (blue), if the range of a that is available during training is constrained. In red the performance on the local test data set D1cl = {D ⊂ D1c|TP1 | a, ∈ t}, which only contains process conditions that are contained in the constrained training data set, is given. In order to get the constrained training data sets, different subsets are sampled from the entire training data set. On the x-axis, the ranges of the absolute time delays a,, which are contained in the used subset of the training data set, normalized by the maximum range of the entire data set D1v|TP1 are given. Since not only the range of the available a, but also their location influences the result, subsets with equal range are sampled at ten different locations each. The result of each used subset is represented by a blue cross mark for the performance on the entire test data set and a red dot mark for the performance on the local test data set. The solid lines represent the respective average values and the shaded area represents the one-sigma range, both calculated based on the results of the ten subsets per given range. The LS- and the WM-trained regression are combined with the linear model without offset and with offset, respectively. For each method, the features are selected by the ISS with s = 100.

# 6.6.6 Domain adaption to different media and attenuation characteristics

In the previous section, the generalizability to other process conditions was discussed. However, another interesting question is whether the training using the data of a specific pipe and medium can also be transferred to other media or to other attenuation characteristics. For example, the medium can be changed to ethanol or a damping mat can be installed. In the field of machine learning, this process of transferring the trained model to other domains—different media or attenuation situations in this thesis—is called domain adaption. Usually, the trained parameters are adapted or fine-tuned to the new domain by considering a small set of labeled measurement signals of the target domain. Since this would require further calibration measurements, which are not desirable, this section investigates the extent to which an already trained FRM can be applied to other media and damping properties *without* fine-tuning or parameter adaptation.

To this end, Figure 6.20 shows the MAPEs of the unsupervised FRM that result when different combinations of the available training and test data sets are used. Note that for the training data sets, only the varyingflow data sets, and for the test data sets only the constant-flow data sets are used, since varying time-delay differences have to be present in the training data sets. Due to there being three varying-flow data sets and three constant-flow data sets, nine different combinations of training and test data sets exist. Because the previous investigation only considered the water data sets without a damping mat, the necessary initial feature set may deviate. Therefore, the results are given for different AWPT tree levels.

It is easy to realize that a transfer from water to ethanol and vice versa is not possible (see the bottom row and the right column of Fig. 6.20). In contrast to that, regardless of whether the FRM is trained on the water data set with or without a damping mat, it can be applied to all water data sets with similar quality. The explanation for this behavior is that the different SoS means that the learned time ranges cannot be transferred to other media. The fact that the unsupervised FRM still shows significantly lower MAPEs for ethanol than the best SotA method (10.8 %) shows that

Figure 6.20 Domain adaptability of the unsupervised FRM, using the ISS (s = 100) and the WM-trained linear regression model (5.35a). Here, the MAPE values of the different TPs using the initial feature sets of the 4th to 9th AWPT tree levels are investigated. In order to test the transferability of the training to situations with different media or attenuation characteristics, all available training data sets are combined with all available test data sets. After training, the FRM is *not* fine-tuned to adapt to the other domain, i.e., it is evaluated to which extent the parameters trained on a specific data set can be applied for the TDE of a data set with a different medium or attenuation characteristic. For the baseline, the MAPE values, when test and training are performed on the same medium and with the same attenuation characteristic, are also included and shown on the diagonal. The data set identifiers can be found in Table 6.1.

the proposed method also works for other media—provided it is also trained on the other medium.

Another interesting parameter to observe is the AWPT-tree level used to generate the initial feature set. For the present ultrasonic system with its sampling rate and bandwidth, the AWPT-tree level does not have too much influence on the results as long as it is chosen smaller than 9. If the level is chosen too high, the resulting basis function will be too extended in the time domain, which will prevent the resolution of different propagation paths.

The last point to notice is the MAPE of the combination D2v/D2c, i.e., using water as medium in a pipe with a damping mat installed. Comparison with the SotA shows that the FRM cannot further improve the systematic error. One possible explanation is the complexity of this data set. Even though the damping mat reduces the interfering signals, the temperature-dependent damping leads to a violation of the assumption that the interfering signals are stationary. Therefore, a regression with constant parameters cannot adequately estimate the delay differences over the wide temperature range of the test data set.

# 6.7 Estimator performances compared with the Cramér-Rao lower bound

The errors in TDE can be separated into systematic errors and stochastic errors. While the systematic errors arise due to wrong model assumptions, interfering signals, or calibration errors, the stochastic errors originate from measurement noise, jitter, or in this particular case, flow fluctuations since the flow is fully turbulent. The systematic errors, which in the considered UFM scenario are mainly caused by the interfering signals, have been extensively studied and compared to the SotA in the previous sections by determination of the MAPEs. Therefore, this section focuses on the stochastic errors of the SDM and FRM. To this end, the stochastic errors of the respective best SDM and FRM are compared to the CRLB—a theoretical lower bound for the variance of unbiased estimators. Note that the CRLB is a valid theoretical lower bound only if the estimators are unbiased and the signal properties such as PSD and observed time range are accurately known. Furthermore, it only represents

a lower limit for stochastic errors that arise due to measurement noise and not due to flow fluctuations. Nevertheless, the CRLB can be used as a reference point to evaluate the magnitude of the stochastic errors in the UFMs. For comparison, the obtained stochastic errors of two SotA methods are also examined.

As the input of the SDM and the FRM are measurement signals with the aim to estimate a time-delay difference, both methods can be considered time delay estimators. Although they are not unbiased due to the existence of interfering signals, the stochastic error can nevertheless be assessed by removing the systematic error beforehand. Similarly, the ZCM and the CC method also represent time delay estimators that are biased due to the interfering signals and whose stochastic errors can be compared to the CRLB. These two methods are specifically chosen because they are representative of a time-based TDE and a correlation-based TDE, respectively.

For the investigation of the stochastic error, artificial AWGN of different levels is added to the measurement signals of the data set D1c|TP1. The level of the artificial AWGN is gradually increased in 20 steps to reduce the initially available SNR stepwise from about 85 dB to about 63 dB. Note that the actual SNR slightly differs for each measurement, since only the amplification of the AWGN can be specified and therefore the actual signal energy of the AWGN is random around its specified mean value. After applying each method under test to the data set at each noise level, the stochastic errors are calculated via removing the systematic errors with the median-based lowpass similar to (6.10). The only difference is that the errors are not processed as percentages but as absolute errors a [] = ̂ − , leading to the standard deviation of the stochastic error ( ̂ − ).

The resulting stochastic errors depending on the SNR are depicted in Fig. 6.21 on the left side. The time ranges used in the ZCM and CC are calculated according to the algorithm described at the end of Sec. 6.3. For the FRM, which is trained on D1v|TP1, the hyperparameters are as follows: ISS (s = 100) for the feature selection, approximated labels for the training, WM-trained linear model with offset for the regression. Furthermore, the results of the SDM in its JBS-based global compensation form are plotted. It is easy to observe that the ZCM and the CC method

Figure 6.21 Considerations on the standard deviation of the proposed estimation methods compared with two SotA methods. As a reference, the results were plotted against the CRLB for the present TDE problem.

perform equally well at high SNR, but the CC method can still be used at low SNR without much degradation. Since the SDM is essentially the same as the ZCM only with a preprocessing step that removes the estimated interfering signals, the stochastic errors are determined by the stochastic components in the estimated interfering signals and the original stochastic errors of the ZCM. From the fact that the SDM and the ZCM perform equally it can be shown that the interfering signals compensation does not add any stochastic components to the measurement signals. The second proposed method, the FRM, can even significantly improve the stochastic error. This can be explained by two properties of the approach. On the one hand, the feature selection leads to a timefrequency selection mask that considers in sum a larger time range for the estimation of the time-delay differences. On the other hand, the combination of several time ranges also achieves an averaging effect over different propagation paths, which also reduces the influence of the flow fluctuation.

Since the different methods use different time ranges of the measurement signal, the CRLB is also different according to (6.11). Therefore, the results are shown again as a function of the CRLB in Fig. 6.21 on the right side. This plot underlines that the FRM not only performs best in terms of stochastic errors but also has the smallest distance to the CRLB, which

can be seen from the fact that for a fixed SNR, the absolute distance to the dashed line is the smallest. The other investigated methods have essentially the same Cramér-Rao efficiency.

# 6.8 Comparison and discussion

All presented methods for TDE (including the SotA) differ in their requirements, such as labeled or unlabeled training sets, signal packages with sufficient signal dynamics, or simply a reference and delay signal. Furthermore, the many hyperparameters of the methods, which all lead to varying results, make an objective and direct comparison of the methods difficult. Therefore, the methods are compared with respect to their requirements: the number of hyperparameters, the necessary preprocessing steps, the achievable measurement dynamics, the practicality, and the results. Finally, it is discussed which method is considered the best overall in the example application presented—the UFM scenario.

Firstly, the requirements are discussed. The SotA methods used as baselines in this dissertation usually only require a single measurement consisting of a reference and delay signal. In contrast to that, the SDM requires a signal package consisting of multiple measurements with varying signal dynamics, i.e., the absolute time delay, the time-delay difference, or both have to vary within the signal package. Then, the timedelay difference can be estimated for each signal within the package. Similarly, the FRM requires such a signal package for the training, for which even the ground truth for the time-delay differences must be known as labels in the case of the supervised FRM.

These requirements on the available measurement signals primarily influence the practicality and the measurement dynamics of a measurement system based on the respective algorithm. The SotA methods, the FRM and the SDMs in global compensation form only need one measurement for the actual TDE, and thus the measurement dynamics are not limited by the algorithm. In contrast, the SDMs in local compensation form or in direct estimation form can only provide estimates after all signals of a signal package have been recorded. This delays the availability of an estimation result on average by half of the time needed to record a signal package with sufficient signal dynamics. Since the signal dynamics

are purely process-driven, this can sometimes be very long. Although the FRM and the SDMs in global compensation form can estimate a timedelay difference individually for each measurement, and thus do not limit the measurement dynamics, they both require a training data set beforehand for training in the case of the FRM and for the globally valid estimation of the interfering signals in case of the SDM. Therefore, in practice, a calibration procedure would be performed before use. While the unsupervised FRM and the SDM can be calibrated without knowledge of the ground truth, the supervised FRM needs the ground truth, which can be provided either by another measuring device or by preparing the time-delay differences in a calibration environment. However, both calibration methods are costly, which is why the unsupervised FRM and the SDMs are preferable in practice.

The next distinguishing feature is the number of hyperparameters. The FRM has the most hyperparameters of all methods, because the AWPT, the feature selection, the regression model and its training strategy must be defined in order to uniquely define the FRM. Nevertheless, the investigations have shown that only certain combinations of hyperparameters lead to very good results, and many hyperparameters, such as the number of features have little effect on quality if they are chosen within certain limits. This can significantly reduce the number of possible hyperparameter combinations. In contrast, the ZCM has only two parameters, namely which zero crossing is to be used and the time range used for the linear approximation of the phase signal. Between the ZCM and the FRM, the rest of the methods fall in line, with the correlation-like SotA methods having more hyperparameters than the ZC method—namely the window type and the processor—and fewer than the SDMs. The SDMs can also be differentiated even more finely. Here, the JBS method has clearly more hyperparameters since the optimization for the spline fit is more complex.

Finally, the quality of the estimates is compared in terms of systematic and stochastic errors. For this purpose, it is assumed that the basic requirements of all methods are met. Then, the supervised FRM leads to the smallest systematic error on average, followed by the unsupervised FRM and the SDM in its global compensation form. The SotA methods all yield approximately similar systematic errors, which are significantly

worse than the SDMs and the FRMs on all data sets. However, this is to be expected, as they are not specifically designed to suppress or filter interfering signals. While the FRM is a separate learning-based approach, those SDMs based on the estimation and subtraction of the interfering signals need to be combined with the SotA methods to perform the TDE. Although the SDMs are combined only with the ZCM in this dissertation, the combination with the other correlation-like methods is also possible, but since the results are expected to be similar, this study was not performed. Regarding the stochastic error, the results show that, again, the FRM performs best and the SDM neither improves nor worsens the SotA methods.

In summary, the unsupervised FRM and the SDM with global compensation are considered to be the best methods because the training data can be collected during normal operation, the measurement dynamics is not limited and the interfering signals are significantly suppressed without the stochastic error becoming worse.

Here, it should be noted briefly that since the successful application of a learning-based approach to TDE, such as the FRM, suggests that TDE can also be accurately performed using approaches from deep learning, various network architectures were tested in a Bachelor thesis [150] to determine the extent to which they can be applied for the present problem. However, the thesis has shown that due to the lacking training data even the deep neural network trained in a supervised fashion is not comparable to the unsupervised FRM, unless a lot of time and effort is invested into developing an adapted network architecture and a targeted augmentation strategy. For these reasons, deep learning approaches were not considered in this dissertation.

# 7 Conclusion

## 7.1 Summary

In this thesis, two TDE techniques for ultrasonic measurement systems have been presented, which are specifically designed to filter or be robust against interfering signals in scenarios where very small time-delay differences have to be estimated with high accuracy. Since these challenges arise in transit-time UFM systems, where the objective is to accurately measure the VoF, the experimental evaluation of the proposed TDE methods is based on a transit-time UFM setup. For the quantitative evaluation, the VoFs in the experiments are converted into the time-delay differences that are to be estimated from the ultrasonic measurement signals. Based on these ground truths of the time-delay differences, the proposed methods can be evaluated in terms of systematic and stochastic error.

Firstly, a signal model has been presented that describes the TDE problem and the origin of the interfering signals in transit-time UFMs. Both presented approach classes—the SDM and the FRM—are based on this signal model, although their principles differ, as the FRM is a learning-based method for the interference-invariant estimation of the time-delay differences and the SDM is a method to estimate and remove the interfering signals before applying a SotA method for TDE.

The requirements of the approaches are that multiple measurements with varying time delays are available, where each measurement signal needs to have a small bandwidth. Furthermore, the additive interfering signals have to be stationary, i.e., they do not vary between consecutive measurements, and symmetric, i.e., they are additively superimposed equally on both the reference and delay signals. Evidence that these requirements are met has been provided by a system identification, which also quantitatively evaluates the extent to which the assumptions of stationarity and symmetry hold. It has been shown that the amplitude and the time delay of the interfering signals are slightly temperaturedependent, which limits the accuracy of the interfering signals estimation, and thus the achievable reduction of the systematic errors.

For the separation of the interfering signals, the SDM employs the principle that the time delays of the target signals vary when multiple measurements are considered, while the interfering signals are stationary. Therefore, this class of methods represents a generalization of the methods proposed by Roosnek [99], where it was assumed that the time delays in the considered measurement package uniformly cover one period of the fundamental oscillation. The proposed SDM can significantly mitigate the boundary condition, so that the time delays can be arbitrarily distributed within the measurement package and the variation of the time delays only needs to be large enough to outweigh the measurement noise. Within the scope of this work, different variants of the SDM have been presented, with the JBS-based global compensation performing the best in the exemplary UFM scenario. The JBS-based SDM in its global compensation form reduced the systematic error on all considered data sets to less than half compared to the best SotA methods. This leads to the conclusion that although the interfering signals could not be completely removed due to their minor non-stationarity, they can be significantly reduced using the proposed SDM.

The FRM uses the AWPT to get a complex-valued time-frequency representation of the measurement signals, which is subsequently used to generate features that correlate with the time delay. After an adapted feature selection, the remaining features are used as input for a regression to directly estimate the time delay differences. The experimental results have shown that the FRM works on different media and is always better than the best SotA method both in terms of systematic and stochastic error. Especially for the stochastic errors, the method is closest to the CRLB. With regard to the dependence on the available training data, the FRM has been shown to estimate the time-delay difference very well even outside the process conditions with which it was trained. In addition, an extension of the FRM has been proposed that also allows an unsupervised training, i.e., the ground truth of the labels for the training data set does not need to be known. Thus, the learning-based approach is directly comparable to the SotA method in terms of practicality, since

a measurement system can easily be adapted to store past measurement signals.

In summary, the objective to reduce the influence of interfering signals in ultrasonic TDEs using the UFM as the application example was achieved. The residual estimation errors of the proposed methods in the experimental results can be explained by the deviation of the signals from the assumed signal model in case of the SDM and by the constraints imposed on the regression model in the case of the FRM.

# 7.2 Outlook

The presented SDM puts a strong focus on the representation of the signal dynamics as point clouds and their processing. However, this limits the approach to the fact that only variable time delays can be used. Using an alternative approach as a Bayesian estimation problem, where the dependence of amplitudes and time delays on process conditions are learned as functions, the interfering signals estimation can then be generalized to determine those interfering signals which have the highest probability, given an observed signal package.

In contrast, the FRM is limited to small time delays due to the feature design—where small means that the transit time is small relative to the period duration of the ultrasonic transducer's operating frequency. This limitation can be overcome by defining other features that combine multiple scales in the AWPT tree to be unambiguous over larger transit times. This in turn requires abandoning the restriction to linear regressions, so other regression types have to be considered in the extended case.

Apart from improving the proposed methods, there are three directions for further research: the application of the proposed TDE algorithm to other applications such as the ultrasonic liquid level metering, the implementation of the improved path-specific TDE for flow profile imaging in UFMs and the extension of the FRM to other machine learning concepts.

While the application to other ultrasonic measurement tasks falls into the category of further validation of the methods, and flow profile imaging is an extension of UFM, from a signal processing perspective, the extension of the methodology to modern learning techniques is most

interesting. Here, different neural network architectures can be explored that can efficiently estimate a time delay on one hand and reduce the influence of interfering signals on the other hand. For this purpose, concepts have to be developed on how to reduce the requirements on the amount of training data as well as on how to design suitable augmentation strategies.

# Bibliography

# List of publications

# List of supervised theses

[160] Wilk, K. *Signalangepasste Zeit-Frequenz-Analyse von Ultraschallsignalen mittels Wavelet-Packets*. Bachelor thesis. 2019.

### **Forschungsberichte aus der Industriellen Informationstechnik (ISSN 2190-6629)**

Die Bände sind unter www.ksp.kit.edu als PDF frei verfügbar oder als Druckausgabe bestellbar.

Die Bände sind unter www.ksp.kit.edu als PDF frei verfügbar oder als Druckausgabe bestellbar.

Gedruckt auf FSC-zertifiziertem Papier

Model-based Filtering of Interfering Signals